{ "license": "CC BY 4.0 — https://creativecommons.org/licenses/by/4.0/", "attribution": "Wheel Cue (wheelcue.com)", "source": "https://wheelcue.com/research/raffle-odds-what-more-tickets-buy/", "slug": "raffle-odds-what-more-tickets-buy", "generatedAt": "2026-10-04T14:14:16.840Z", "node": "v24.21.0", "randomJsSha256": "946e82a3c426d698779d3246c37c732f771db3873c93239ef3952aea39be1e2d", "totalSpins": 41500000, "rawDraws": 41500001, "seconds": 103.9, "checksTotal": 97, "checksPassed": 97, "params": { "onePrize": { "pools": [ 100, 300, 1000 ], "alsoSimulatedPools": [ 101, 301 ], "draws": 1000000, "ticketsHeld": [ 1, 2, 5, 10, 20 ], "bigBook": { "tickets": 1000, "blocks": 10, "ticketsPerBlock": 100, "note": "drawn as block (1 of 10) then ticket (1 of 100): one wheel holds at most 500 entries", "secondPrizeSessions": 500000, "secondPrizeProcedure": "block wheel with Weighted choices on and retirement off, lines 'Block k *100'; after the first prize the winning block's line becomes '*99' and its winning ticket is left off when that block's tickets are loaded again" } }, "severalPrizes": { "rule": "raffle wheel, each ticket retired once it wins", "tickets": 300, "prizes": 10, "prizeCounts": [ 1, 3, 5, 10 ], "ticketsHeld": [ 1, 2, 5, 10, 20, 30 ], "sessions": 500000, "chartTicketsMax": 60 }, "onePerPerson": { "rule": "giveaway picker, the winner's whole line (all entries) removed", "others": 99, "prizes": 5, "entries": [ 1, 2, 3, 5, 10 ], "retiredEntriesSimulated": [ 5, 10 ], "theoryOnlyEntries": [ 20 ], "sessions": 600000 }, "mix": { "people": [ { "entries": 1, "people": 70 }, { "entries": 3, "people": 20 }, { "entries": 10, "people": 10 } ], "prizes": 5, "sessions": 600000 }, "bonus": { "people": 200, "baseEntries": 1, "bonusEntries": 4, "prizes": 3, "sharersTable": [ 1, 20, 50, 100, 150, 200 ], "chartOthersSharing": { "from": 0, "to": 199, "step": 5 }, "simulatedSharers": 50, "sessions": 500000 } }, "notes": { "simulatedVsExact": "results holds simulated draws (never site usage); theory holds exact math computed with exact fractions (BigInt) and rounded to 6 decimals. Quote theory values; call results values 'simulated'.", "siteBehavior": "Raffle wheel: 'Retire each ticket once it wins' is ON by default (one prize per ticket; a holder of several tickets can win several prizes). Giveaway winner picker: 'Remove each winner before the next draw' is ON by default; Weighted choices is OFF by default on both tools, and with it off a line 'Sam *3' is one entry named 'Sam *3' with weight 1, so it must be turned on before writing *3. With it on, a line 'Sam *3' is removed whole when it wins, so a person wins at most once. A person pasted on several plain lines loses only one line per win, which silently becomes the ticket rule (several prizes possible): for one prize per person, use one weighted line per person, never duplicates. One wheel takes at most 500 entries. Neither wheel expands a typed range: '101-150' stays one entry, so never tell readers to type a range. The raffle wheel's FAQ points to the random number generator to write the numbers out, and that tool caps a range at 200 numbers, so a 300-ticket book takes two batches (e.g. 1-200, then 201-300) pasted into one list. For a big book the raffle wheel page suggests drawing a block first, then the ticket; see theory.onePrize.bigBookAfterRetirement before using that for more than one prize.", "spins": "totalSpins counts CueRandom.pickWeighted() calls, one per wheel spin (the 1,000-ticket draw is two spins: block, then ticket). rawDraws counts 32-bit numbers from crypto.getRandomValues(), including the few the rejection step discards.", "checks": "Each check: z = (simulated - exact)/se, pass when |z| <= 4. Uniformity checks compare a chi-square statistic with its expected value (df), se = sqrt(2 df). A group's mean prize count uses the sample SD of that count across sessions. A per-person share pooled over the people of a group uses the sample SD of the per-session share (people within a session are not independent; sessions are). One-ticket expected-prize and non-sharer group checks are left out because they would repeat another check exactly.", "rounding": "display strings: two decimals below 10% or above 90%, one decimal between; 'points' are percentage points with two decimals." }, "results": { "onePrize": { "note": "pWinHoldingFirstT[t] = simulated share of draws won by tickets 1..t (every ticket is alike, so it is the chance of a holder of t tickets). 101 and 301 are the pools after one more ticket is sold. mostWinsForOneTicket/fewestWinsForOneTicket are from this one run and change on every run.", "pools": [ { "tickets": 100, "method": "one wheel", "draws": 1000000, "chiSquare": 93.5, "df": 99, "fewestWinsForOneTicket": 9728, "mostWinsForOneTicket": 10239, "pWinHoldingFirstT": { "1": 0.010102, "2": 0.020311, "3": 0.030229, "5": 0.050137, "6": 0.060244, "10": 0.100357, "11": 0.110381, "20": 0.20006, "21": 0.210072 } }, { "tickets": 101, "method": "one wheel", "draws": 1000000, "chiSquare": 102.9, "df": 100, "fewestWinsForOneTicket": 9655, "mostWinsForOneTicket": 10168, "pWinHoldingFirstT": { "1": 0.009793, "2": 0.019667, "3": 0.029636, "5": 0.049373, "6": 0.059272, "10": 0.098661, "11": 0.108541, "20": 0.197926, "21": 0.207747 } }, { "tickets": 300, "method": "one wheel", "draws": 1000000, "chiSquare": 300, "df": 299, "fewestWinsForOneTicket": 3193, "mostWinsForOneTicket": 3520, "pWinHoldingFirstT": { "1": 0.003332, "2": 0.006728, "3": 0.010062, "5": 0.016788, "6": 0.020102, "10": 0.033295, "11": 0.036525, "20": 0.066264, "21": 0.069593 } }, { "tickets": 301, "method": "one wheel", "draws": 1000000, "chiSquare": 322.1, "df": 300, "fewestWinsForOneTicket": 3165, "mostWinsForOneTicket": 3511, "pWinHoldingFirstT": { "1": 0.00331, "2": 0.006633, "3": 0.009915, "5": 0.016502, "6": 0.019773, "10": 0.03309, "11": 0.03633, "20": 0.066228, "21": 0.069555 } }, { "tickets": 1000, "method": "block (1 of 10) then ticket (1 of 100)", "draws": 1000000, "chiSquare": 1003.5, "df": 999, "fewestWinsForOneTicket": 893, "mostWinsForOneTicket": 1098, "pWinHoldingFirstT": { "1": 0.00102, "2": 0.002018, "3": 0.003065, "5": 0.005045, "6": 0.006097, "10": 0.01006, "11": 0.011113, "20": 0.020234, "21": 0.021264 } } ], "bigBookSecondPrize": { "sessions": 500000, "procedure": "block wheel with Weighted choices on and retirement off, lines 'Block k *100'; after the first prize the winning block's line becomes '*99' and its winning ticket is left off when that block's tickets are loaded again", "sameBlockAgain": 0.098448, "chiSquare": 1031.7, "df": 998, "note": "Simulated: first prize by block then ticket, second prize with the winning block weighted 99 against 100 and its winning ticket left out. sameBlockAgain = share of sessions in which the second prize came from the first winner's block." } }, "severalPrizes": { "tickets": 300, "sessions": 500000, "prizesPerSession": 10, "note": "Each session draws 10 prizes from 300 tickets with every winning ticket retired; the first m draws of a session are an m-prize raffle. The holder of t tickets holds tickets 1..t.", "table": [ { "prizes": 1, "yourTickets": 1, "pAtLeastOne": 0.003448, "pAtLeastTwo": 0, "meanPrizes": 0.003448 }, { "prizes": 1, "yourTickets": 2, "pAtLeastOne": 0.006728, "pAtLeastTwo": 0, "meanPrizes": 0.006728 }, { "prizes": 1, "yourTickets": 5, "pAtLeastOne": 0.016884, "pAtLeastTwo": 0, "meanPrizes": 0.016884 }, { "prizes": 1, "yourTickets": 10, "pAtLeastOne": 0.033566, "pAtLeastTwo": 0, "meanPrizes": 0.033566 }, { "prizes": 1, "yourTickets": 20, "pAtLeastOne": 0.066674, "pAtLeastTwo": 0, "meanPrizes": 0.066674 }, { "prizes": 1, "yourTickets": 30, "pAtLeastOne": 0.099916, "pAtLeastTwo": 0, "meanPrizes": 0.099916 }, { "prizes": 3, "yourTickets": 1, "pAtLeastOne": 0.010208, "pAtLeastTwo": 0, "meanPrizes": 0.010208 }, { "prizes": 3, "yourTickets": 2, "pAtLeastOne": 0.020048, "pAtLeastTwo": 0.000078, "meanPrizes": 0.020126 }, { "prizes": 3, "yourTickets": 5, "pAtLeastOne": 0.04965, "pAtLeastTwo": 0.000706, "meanPrizes": 0.050356 }, { "prizes": 3, "yourTickets": 10, "pAtLeastOne": 0.097282, "pAtLeastTwo": 0.002996, "meanPrizes": 0.100306 }, { "prizes": 3, "yourTickets": 20, "pAtLeastOne": 0.18775, "pAtLeastTwo": 0.012206, "meanPrizes": 0.20019 }, { "prizes": 3, "yourTickets": 30, "pAtLeastOne": 0.271334, "pAtLeastTwo": 0.027384, "meanPrizes": 0.299638 }, { "prizes": 5, "yourTickets": 1, "pAtLeastOne": 0.016706, "pAtLeastTwo": 0, "meanPrizes": 0.016706 }, { "prizes": 5, "yourTickets": 2, "pAtLeastOne": 0.033068, "pAtLeastTwo": 0.000202, "meanPrizes": 0.03327 }, { "prizes": 5, "yourTickets": 5, "pAtLeastOne": 0.081378, "pAtLeastTwo": 0.002266, "meanPrizes": 0.083666 }, { "prizes": 5, "yourTickets": 10, "pAtLeastOne": 0.157294, "pAtLeastTwo": 0.00937, "meanPrizes": 0.166968 }, { "prizes": 5, "yourTickets": 20, "pAtLeastOne": 0.293738, "pAtLeastTwo": 0.03718, "meanPrizes": 0.333314 }, { "prizes": 5, "yourTickets": 30, "pAtLeastOne": 0.411144, "pAtLeastTwo": 0.079712, "meanPrizes": 0.4993 }, { "prizes": 10, "yourTickets": 1, "pAtLeastOne": 0.033482, "pAtLeastTwo": 0, "meanPrizes": 0.033482 }, { "prizes": 10, "yourTickets": 2, "pAtLeastOne": 0.065586, "pAtLeastTwo": 0.000992, "meanPrizes": 0.066578 }, { "prizes": 10, "yourTickets": 5, "pAtLeastOne": 0.156582, "pAtLeastTwo": 0.009644, "meanPrizes": 0.166476 }, { "prizes": 10, "yourTickets": 10, "pAtLeastOne": 0.291456, "pAtLeastTwo": 0.039086, "meanPrizes": 0.333536 }, { "prizes": 10, "yourTickets": 20, "pAtLeastOne": 0.50359, "pAtLeastTwo": 0.137964, "meanPrizes": 0.66659 }, { "prizes": 10, "yourTickets": 30, "pAtLeastOne": 0.65693, "pAtLeastTwo": 0.262768, "meanPrizes": 0.9992 } ], "holderOf10ByDraw": [ { "draw": 1, "simulated": 0.033566 }, { "draw": 2, "simulated": 0.03362 }, { "draw": 3, "simulated": 0.03312 }, { "draw": 4, "simulated": 0.033254 }, { "draw": 5, "simulated": 0.033408 }, { "draw": 6, "simulated": 0.03339 }, { "draw": 7, "simulated": 0.033832 }, { "draw": 8, "simulated": 0.033154 }, { "draw": 9, "simulated": 0.033402 }, { "draw": 10, "simulated": 0.03279 } ] }, "onePerPerson": { "others": 99, "prizes": 5, "note": "'person' = Weighted choices on and the person on one line 'You *w'; the whole line leaves when it wins (giveaway picker default). 'tickets' = the person pasted on w plain lines; removal takes only one line per win, which is the raffle-wheel rule (retired tickets). wonDraw[j] = simulated share of sessions in which this person won the prize drawn (j+1)-th.", "rows": [ { "entries": 1, "rule": "person", "sessions": 600000, "wonDraw": [ 0.010258, 0.009853, 0.010162, 0.010048, 0.010088 ], "pAny": 0.05041 }, { "entries": 2, "rule": "person", "sessions": 600000, "wonDraw": [ 0.020067, 0.019777, 0.0196, 0.018975, 0.01921 ], "pAny": 0.097628 }, { "entries": 3, "rule": "person", "sessions": 600000, "wonDraw": [ 0.029283, 0.029053, 0.028123, 0.027337, 0.02677 ], "pAny": 0.140567 }, { "entries": 5, "rule": "person", "sessions": 600000, "wonDraw": [ 0.048098, 0.046278, 0.04411, 0.042622, 0.04081 ], "pAny": 0.221918 }, { "entries": 10, "rule": "person", "sessions": 600000, "wonDraw": [ 0.091682, 0.0836, 0.077455, 0.070177, 0.064697 ], "pAny": 0.38761 }, { "entries": 5, "rule": "tickets", "sessions": 600000, "wonDraw": [ 0.047752, 0.048177, 0.048305, 0.048117, 0.048527 ], "pAny": 0.222552, "pTwoOrMore": 0.017783, "meanPrizes": 0.240877 }, { "entries": 10, "rule": "tickets", "sessions": 600000, "wonDraw": [ 0.091112, 0.091197, 0.091383, 0.091123, 0.091922 ], "pAny": 0.386172, "pTwoOrMore": 0.065217, "meanPrizes": 0.456737 } ] }, "mix": { "entries": 230, "people": 100, "prizes": 5, "sessions": 600000, "note": "shareOfDraw[j] = share of sessions in which the prize drawn (j+1)-th went to someone in this group. perPerson values are group values divided by the group size. The pooled per-person checks (ticket rule, at least one / two or more) take one value per session, the share of the group's people who reached it, and use that value's sample SD across sessions: the people in one session are not independent, the sessions are.", "groups": [ { "entries": 1, "people": 70, "person": { "shareOfDraw": [ 0.303532, 0.310958, 0.316598, 0.323077, 0.330668 ], "perPersonByDraw": [ 0.004336, 0.004442, 0.004523, 0.004615, 0.004724 ], "perPersonAny": 0.02264, "groupMeanPrizes": 1.584833 }, "tickets": { "shareOfDraw": [ 0.304607, 0.30425, 0.30461, 0.305185, 0.304823 ], "perPersonAny": 0.021764, "perPersonTwoOrMore": 0, "groupMeanPrizes": 1.523475 } }, { "entries": 3, "people": 20, "person": { "shareOfDraw": [ 0.2618, 0.262933, 0.267763, 0.269103, 0.272 ], "perPersonByDraw": [ 0.01309, 0.013147, 0.013388, 0.013455, 0.0136 ], "perPersonAny": 0.06668, "groupMeanPrizes": 1.3336 }, "tickets": { "shareOfDraw": [ 0.261733, 0.260557, 0.260217, 0.261762, 0.260283 ], "perPersonAny": 0.064095, "perPersonTwoOrMore": 0.001128, "groupMeanPrizes": 1.304552 } }, { "entries": 10, "people": 10, "person": { "shareOfDraw": [ 0.434668, 0.426108, 0.415638, 0.40782, 0.397332 ], "perPersonByDraw": [ 0.043467, 0.042611, 0.041564, 0.040782, 0.039733 ], "perPersonAny": 0.208157, "groupMeanPrizes": 2.081567 }, "tickets": { "shareOfDraw": [ 0.43366, 0.435193, 0.435173, 0.433053, 0.434893 ], "perPersonAny": 0.200667, "perPersonTwoOrMore": 0.015938, "groupMeanPrizes": 2.171973 } } ] }, "bonus": { "people": 200, "sharers": 50, "entriesPerSharer": 5, "prizes": 3, "sessions": 500000, "sharersGroupByDraw": [ 0.623734, 0.623178, 0.62042 ], "sharerAny": 0.037347, "nonSharerAny": 0.007551, "oneParticularNonSharerAny": 0.007574, "note": "One prize per person (whole line removed). sharerAny = simulated prizes won by the sharers' group / number of sharers; nonSharerAny likewise (it equals (3 - the sharers' prizes)/150, so it carries no extra evidence). oneParticularNonSharerAny = share of sessions in which one specific non-sharer won a prize (a separate, noisier estimate of the same exact value)." } }, "theory": { "onePrize": { "formulas": { "p": "t/N (each of N tickets equally likely)", "pAfterOneMore": "(t+1)/(N+1): buying one more ticket also adds it to the pool", "gain": "(t+1)/(N+1) - t/N = (N-t)/(N(N+1)), always less than 1/N; gainVsOneOverN = gain × N = (N-t)/(N+1), close to 1 when your share t/N is small and smaller as it grows", "pDoubled": "2t/(N+t): you buy t more tickets and the pool grows by t; doubledRatio = 2N/(N+t) < 2" }, "note": "With one prize the chance is in exact proportion to your share of the tickets, t/N. When buying a ticket also adds it to the pool, each extra ticket adds (N-t)/(N(N+1)): close to 1/N while your share is small, and noticeably less as it grows (gainVsOneOverN: 0.79 × 1/N at 20 of 100 tickets, 0.93 × 1/N at 20 of 300). Quote gain or gainVsOneOverN from the rows, not a rule of thumb. The 1,000-ticket raffle was simulated as a two-step draw (block, then ticket) because one wheel holds at most 500 entries; with equal blocks of 100 and no ticket retired yet, every ticket has exactly 1/10 × 1/100 = 1/1,000.", "bigBookAfterRetirement": { "note": "Exact arithmetic for the second prize of a 1,000-ticket book drawn block-then-ticket, after the first prize went to a ticket in one block (99 of its tickets left, 900 in the other blocks). A fair draw gives each of the 999 remaining tickets 1/999. 'retiredBlock' is what the raffle wheel does by default ('Retire each ticket once it wins' is on, so the winning block's line leaves the block wheel): its other 99 tickets cannot win the second prize. 'equalBlocks' (retirement off, blocks unchanged) gives them 1/990 against 1/1,000. Only 'weightedBlocks' is exact: turn retirement off on the block wheel, turn Weighted choices on, write every block as 'Block k *100' and lower the winning block to '*99' (a weighted line would also be removed whole if retirement were on). That procedure is simulated in results.onePrize.bigBookSecondPrize.", "fairChance": 0.001001, "fairChanceFraction": "1/999", "retiredBlock": { "ticketInWinnersBlock": 0, "ticketInWinnersBlockFraction": "0", "otherTicket": 0.001111, "otherTicketFraction": "1/900", "winnersBlockAgain": 0, "winnersBlockAgainFraction": "0" }, "equalBlocks": { "ticketInWinnersBlock": 0.00101, "ticketInWinnersBlockFraction": "1/990", "otherTicket": 0.001, "otherTicketFraction": "1/1000", "winnersBlockAgain": 0.1, "winnersBlockAgainFraction": "1/10" }, "weightedBlocks": { "ticketInWinnersBlock": 0.001001, "ticketInWinnersBlockFraction": "1/999", "otherTicket": 0.001001, "otherTicketFraction": "1/999", "winnersBlockAgain": 0.099099, "winnersBlockAgainFraction": "11/111" } }, "rows": [ { "tickets": 100, "yourTickets": 1, "p": 0.01, "fraction": "1/100", "display": "1.00%", "oneIn": "1 in 100", "pAfterOneMore": 0.019802, "fractionAfterOneMore": "2/101", "displayAfterOneMore": "1.98%", "gain": 0.00980198, "gainFraction": "99/10100", "gainDisplay": "0.98 points", "gainVsOneOverN": 0.980198, "pDoubled": 0.019802, "displayDoubled": "1.98%", "doubledRatio": 1.9802 }, { "tickets": 100, "yourTickets": 2, "p": 0.02, "fraction": "1/50", "display": "2.00%", "oneIn": "1 in 50", "pAfterOneMore": 0.029703, "fractionAfterOneMore": "3/101", "displayAfterOneMore": "2.97%", "gain": 0.00970297, "gainFraction": "49/5050", "gainDisplay": "0.97 points", "gainVsOneOverN": 0.970297, "pDoubled": 0.039216, "displayDoubled": "3.92%", "doubledRatio": 1.9608 }, { "tickets": 100, "yourTickets": 5, "p": 0.05, "fraction": "1/20", "display": "5.00%", "oneIn": "1 in 20", "pAfterOneMore": 0.059406, "fractionAfterOneMore": "6/101", "displayAfterOneMore": "5.94%", "gain": 0.00940594, "gainFraction": "19/2020", "gainDisplay": "0.94 points", "gainVsOneOverN": 0.940594, "pDoubled": 0.095238, "displayDoubled": "9.52%", "doubledRatio": 1.9048 }, { "tickets": 100, "yourTickets": 10, "p": 0.1, "fraction": "1/10", "display": "10.0%", "oneIn": "1 in 10", "pAfterOneMore": 0.108911, "fractionAfterOneMore": "11/101", "displayAfterOneMore": "10.9%", "gain": 0.00891089, "gainFraction": "9/1010", "gainDisplay": "0.89 points", "gainVsOneOverN": 0.891089, "pDoubled": 0.181818, "displayDoubled": "18.2%", "doubledRatio": 1.8182 }, { "tickets": 100, "yourTickets": 20, "p": 0.2, "fraction": "1/5", "display": "20.0%", "oneIn": "1 in 5", "pAfterOneMore": 0.207921, "fractionAfterOneMore": "21/101", "displayAfterOneMore": "20.8%", "gain": 0.00792079, "gainFraction": "4/505", "gainDisplay": "0.79 points", "gainVsOneOverN": 0.792079, "pDoubled": 0.333333, "displayDoubled": "33.3%", "doubledRatio": 1.6667 }, { "tickets": 300, "yourTickets": 1, "p": 0.003333, "fraction": "1/300", "display": "0.33%", "oneIn": "1 in 300", "pAfterOneMore": 0.006645, "fractionAfterOneMore": "2/301", "displayAfterOneMore": "0.66%", "gain": 0.00331118, "gainFraction": "299/90300", "gainDisplay": "0.33 points", "gainVsOneOverN": 0.993355, "pDoubled": 0.006645, "displayDoubled": "0.66%", "doubledRatio": 1.9934 }, { "tickets": 300, "yourTickets": 2, "p": 0.006667, "fraction": "1/150", "display": "0.67%", "oneIn": "1 in 150", "pAfterOneMore": 0.009967, "fractionAfterOneMore": "3/301", "displayAfterOneMore": "1.00%", "gain": 0.00330011, "gainFraction": "149/45150", "gainDisplay": "0.33 points", "gainVsOneOverN": 0.990033, "pDoubled": 0.013245, "displayDoubled": "1.32%", "doubledRatio": 1.9868 }, { "tickets": 300, "yourTickets": 5, "p": 0.016667, "fraction": "1/60", "display": "1.67%", "oneIn": "1 in 60", "pAfterOneMore": 0.019934, "fractionAfterOneMore": "6/301", "displayAfterOneMore": "1.99%", "gain": 0.00326689, "gainFraction": "59/18060", "gainDisplay": "0.33 points", "gainVsOneOverN": 0.980066, "pDoubled": 0.032787, "displayDoubled": "3.28%", "doubledRatio": 1.9672 }, { "tickets": 300, "yourTickets": 10, "p": 0.033333, "fraction": "1/30", "display": "3.33%", "oneIn": "1 in 30", "pAfterOneMore": 0.036545, "fractionAfterOneMore": "11/301", "displayAfterOneMore": "3.65%", "gain": 0.00321152, "gainFraction": "29/9030", "gainDisplay": "0.32 points", "gainVsOneOverN": 0.963455, "pDoubled": 0.064516, "displayDoubled": "6.45%", "doubledRatio": 1.9355 }, { "tickets": 300, "yourTickets": 20, "p": 0.066667, "fraction": "1/15", "display": "6.67%", "oneIn": "1 in 15", "pAfterOneMore": 0.069767, "fractionAfterOneMore": "3/43", "displayAfterOneMore": "6.98%", "gain": 0.00310078, "gainFraction": "2/645", "gainDisplay": "0.31 points", "gainVsOneOverN": 0.930233, "pDoubled": 0.125, "displayDoubled": "12.5%", "doubledRatio": 1.875 }, { "tickets": 1000, "yourTickets": 1, "p": 0.001, "fraction": "1/1000", "display": "0.10%", "oneIn": "1 in 1,000", "pAfterOneMore": 0.001998, "fractionAfterOneMore": "2/1001", "displayAfterOneMore": "0.20%", "gain": 0.000998002, "gainFraction": "999/1001000", "gainDisplay": "0.10 points", "gainVsOneOverN": 0.998002, "pDoubled": 0.001998, "displayDoubled": "0.20%", "doubledRatio": 1.998 }, { "tickets": 1000, "yourTickets": 2, "p": 0.002, "fraction": "1/500", "display": "0.20%", "oneIn": "1 in 500", "pAfterOneMore": 0.002997, "fractionAfterOneMore": "3/1001", "displayAfterOneMore": "0.30%", "gain": 0.000997003, "gainFraction": "499/500500", "gainDisplay": "0.10 points", "gainVsOneOverN": 0.997003, "pDoubled": 0.003992, "displayDoubled": "0.40%", "doubledRatio": 1.996 }, { "tickets": 1000, "yourTickets": 5, "p": 0.005, "fraction": "1/200", "display": "0.50%", "oneIn": "1 in 200", "pAfterOneMore": 0.005994, "fractionAfterOneMore": "6/1001", "displayAfterOneMore": "0.60%", "gain": 0.000994006, "gainFraction": "199/200200", "gainDisplay": "0.10 points", "gainVsOneOverN": 0.994006, "pDoubled": 0.00995, "displayDoubled": "1.00%", "doubledRatio": 1.99 }, { "tickets": 1000, "yourTickets": 10, "p": 0.01, "fraction": "1/100", "display": "1.00%", "oneIn": "1 in 100", "pAfterOneMore": 0.010989, "fractionAfterOneMore": "1/91", "displayAfterOneMore": "1.10%", "gain": 0.000989011, "gainFraction": "9/9100", "gainDisplay": "0.10 points", "gainVsOneOverN": 0.989011, "pDoubled": 0.019802, "displayDoubled": "1.98%", "doubledRatio": 1.9802 }, { "tickets": 1000, "yourTickets": 20, "p": 0.02, "fraction": "1/50", "display": "2.00%", "oneIn": "1 in 50", "pAfterOneMore": 0.020979, "fractionAfterOneMore": "3/143", "displayAfterOneMore": "2.10%", "gain": 0.000979021, "gainFraction": "7/7150", "gainDisplay": "0.10 points", "gainVsOneOverN": 0.979021, "pDoubled": 0.039216, "displayDoubled": "3.92%", "doubledRatio": 1.9608 } ] }, "severalPrizes": { "tickets": 300, "formulas": { "pAtLeastOne": "1 - C(N-t, m)/C(N, m)", "pAtLeastTwo": "1 - [C(N-t, m) + t·C(N-t, m-1)]/C(N, m)", "pExactly": "C(t, k)·C(N-t, m-k)/C(N, m) (hypergeometric), k = 0, 1, 2, ...", "expectedPrizes": "m·t/N exactly (linearity: each of your t tickets wins one of the m prizes with chance m/N)", "nextTicketExpected": "fixed pool: m(t+1)/N - mt/N = m/N for every t; pool grows: m(t+1)/(N+1) - mt/N = m(N-t)/(N(N+1))", "byDraw": "with tickets retired, a holder of t tickets wins the prize drawn j-th with chance t/N for every j (symmetry), so the draw order does not matter" }, "note": "Computed with exact fractions (BigInt). pExactly lists k = 0, 1, 2, 3 and any larger k whose exact probability is at least 1e-6. chart is for a fixed 300-ticket raffle with 10 prizes, t of the tickets yours. nextTicket: 'fixed pool' = you hold one more of the same 300 tickets (someone else holds one fewer); 'pool grows' = one more ticket is sold, to you (301 tickets). Only in the fixed pool is the addition to expected prizes a constant m/N = 1/30; when the pool grows it is m(N-t)/(N(N+1)), which falls as you hold more (addsToExpectedPoolGrows). Name the pool whenever a 'constant' is quoted. All 'addsTo' fields have 6 significant digits.", "table": [ { "prizes": 1, "yourTickets": 1, "pAtLeastOne": 0.003333, "displayAtLeastOne": "0.33%", "pAtLeastTwo": 0, "displayAtLeastTwo": "0%", "expectedPrizes": 0.003333, "expectedFraction": "1/300", "pExactly": [ 0.996667, 0.003333 ] }, { "prizes": 1, "yourTickets": 2, "pAtLeastOne": 0.006667, "displayAtLeastOne": "0.67%", "pAtLeastTwo": 0, "displayAtLeastTwo": "0%", "expectedPrizes": 0.006667, "expectedFraction": "1/150", "pExactly": [ 0.993333, 0.006667 ] }, { "prizes": 1, "yourTickets": 5, "pAtLeastOne": 0.016667, "displayAtLeastOne": "1.67%", "pAtLeastTwo": 0, "displayAtLeastTwo": "0%", "expectedPrizes": 0.016667, "expectedFraction": "1/60", "pExactly": [ 0.983333, 0.016667 ] }, { "prizes": 1, "yourTickets": 10, "pAtLeastOne": 0.033333, "displayAtLeastOne": "3.33%", "pAtLeastTwo": 0, "displayAtLeastTwo": "0%", "expectedPrizes": 0.033333, "expectedFraction": "1/30", "pExactly": [ 0.966667, 0.033333 ] }, { "prizes": 1, "yourTickets": 20, "pAtLeastOne": 0.066667, "displayAtLeastOne": "6.67%", "pAtLeastTwo": 0, "displayAtLeastTwo": "0%", "expectedPrizes": 0.066667, "expectedFraction": "1/15", "pExactly": [ 0.933333, 0.066667 ] }, { "prizes": 1, "yourTickets": 30, "pAtLeastOne": 0.1, "displayAtLeastOne": "10.0%", "pAtLeastTwo": 0, "displayAtLeastTwo": "0%", "expectedPrizes": 0.1, "expectedFraction": "1/10", "pExactly": [ 0.9, 0.1 ] }, { "prizes": 3, "yourTickets": 1, "pAtLeastOne": 0.01, "displayAtLeastOne": "1.00%", "pAtLeastTwo": 0, "displayAtLeastTwo": "0%", "expectedPrizes": 0.01, "expectedFraction": "1/100", "pExactly": [ 0.99, 0.01 ] }, { "prizes": 3, "yourTickets": 2, "pAtLeastOne": 0.019933, "displayAtLeastOne": "1.99%", "pAtLeastTwo": 0.000067, "displayAtLeastTwo": "<0.01%", "expectedPrizes": 0.02, "expectedFraction": "1/50", "pExactly": [ 0.980067, 0.019866, 0.000067 ] }, { "prizes": 3, "yourTickets": 5, "pAtLeastOne": 0.049333, "displayAtLeastOne": "4.93%", "pAtLeastTwo": 0.000664, "displayAtLeastTwo": "0.07%", "expectedPrizes": 0.05, "expectedFraction": "1/20", "pExactly": [ 0.950667, 0.048669, 0.000662, 0.000002 ] }, { "prizes": 3, "yourTickets": 10, "pAtLeastOne": 0.097017, "displayAtLeastOne": "9.70%", "pAtLeastTwo": 0.002956, "displayAtLeastTwo": "0.30%", "expectedPrizes": 0.1, "expectedFraction": "1/10", "pExactly": [ 0.902983, 0.094061, 0.002929, 0.000027 ] }, { "prizes": 3, "yourTickets": 20, "pAtLeastOne": 0.187547, "displayAtLeastOne": "18.8%", "pAtLeastTwo": 0.012197, "displayAtLeastTwo": "1.22%", "expectedPrizes": 0.2, "expectedFraction": "1/5", "pExactly": [ 0.812453, 0.17535, 0.011941, 0.000256 ] }, { "prizes": 3, "yourTickets": 30, "pAtLeastOne": 0.271814, "displayAtLeastOne": "27.2%", "pAtLeastTwo": 0.027274, "displayAtLeastTwo": "2.73%", "expectedPrizes": 0.3, "expectedFraction": "3/10", "pExactly": [ 0.728186, 0.24454, 0.026363, 0.000911 ] }, { "prizes": 5, "yourTickets": 1, "pAtLeastOne": 0.016667, "displayAtLeastOne": "1.67%", "pAtLeastTwo": 0, "displayAtLeastTwo": "0%", "expectedPrizes": 0.016667, "expectedFraction": "1/60", "pExactly": [ 0.983333, 0.016667 ] }, { "prizes": 5, "yourTickets": 2, "pAtLeastOne": 0.03311, "displayAtLeastOne": "3.31%", "pAtLeastTwo": 0.000223, "displayAtLeastTwo": "0.02%", "expectedPrizes": 0.033333, "expectedFraction": "1/30", "pExactly": [ 0.96689, 0.032887, 0.000223 ] }, { "prizes": 5, "yourTickets": 5, "pAtLeastOne": 0.081126, "displayAtLeastOne": "8.11%", "pAtLeastTwo": 0.002185, "displayAtLeastTwo": "0.22%", "expectedPrizes": 0.083333, "expectedFraction": "1/12", "pExactly": [ 0.918874, 0.078941, 0.002163, 0.000022 ] }, { "prizes": 5, "yourTickets": 10, "pAtLeastOne": 0.156899, "displayAtLeastOne": "15.7%", "pAtLeastTwo": 0.009504, "displayAtLeastTwo": "0.95%", "expectedPrizes": 0.166667, "expectedFraction": "1/6", "pExactly": [ 0.843101, 0.147395, 0.009244, 0.000257, 0.000003 ] }, { "prizes": 5, "yourTickets": 20, "pAtLeastOne": 0.293456, "displayAtLeastOne": "29.3%", "pAtLeastTwo": 0.037462, "displayAtLeastTwo": "3.75%", "expectedPrizes": 0.333333, "expectedFraction": "1/3", "pExactly": [ 0.706544, 0.255994, 0.035118, 0.002274, 0.000069 ] }, { "prizes": 5, "yourTickets": 30, "pAtLeastOne": 0.411716, "displayAtLeastOne": "41.2%", "pAtLeastTwo": 0.079977, "displayAtLeastTwo": "8.00%", "expectedPrizes": 0.5, "expectedFraction": "1/2", "pExactly": [ 0.588284, 0.331739, 0.072063, 0.007529, 0.000378, 0.000007 ] }, { "prizes": 10, "yourTickets": 1, "pAtLeastOne": 0.033333, "displayAtLeastOne": "3.33%", "pAtLeastTwo": 0, "displayAtLeastTwo": "0%", "expectedPrizes": 0.033333, "expectedFraction": "1/30", "pExactly": [ 0.966667, 0.033333 ] }, { "prizes": 10, "yourTickets": 2, "pAtLeastOne": 0.065663, "displayAtLeastOne": "6.57%", "pAtLeastTwo": 0.001003, "displayAtLeastTwo": "0.10%", "expectedPrizes": 0.066667, "expectedFraction": "1/15", "pExactly": [ 0.934337, 0.06466, 0.001003 ] }, { "prizes": 10, "yourTickets": 5, "pAtLeastOne": 0.156899, "displayAtLeastOne": "15.7%", "pAtLeastTwo": 0.009504, "displayAtLeastTwo": "0.95%", "expectedPrizes": 0.166667, "expectedFraction": "1/6", "pExactly": [ 0.843101, 0.147395, 0.009244, 0.000257, 0.000003 ] }, { "prizes": 10, "yourTickets": 10, "pAtLeastOne": 0.291285, "displayAtLeastOne": "29.1%", "pAtLeastTwo": 0.039073, "displayAtLeastTwo": "3.91%", "expectedPrizes": 0.333333, "expectedFraction": "1/3", "pExactly": [ 0.708715, 0.252212, 0.036222, 0.002731, 0.000118, 0.000003 ] }, { "prizes": 10, "yourTickets": 20, "pAtLeastOne": 0.503853, "displayAtLeastOne": "50.4%", "pAtLeastTwo": 0.137693, "displayAtLeastTwo": "13.8%", "expectedPrizes": 0.666667, "expectedFraction": "2/3", "pExactly": [ 0.496147, 0.36616, 0.115098, 0.020237, 0.002197, 0.000153, 0.000007 ] }, { "prizes": 10, "yourTickets": 30, "pAtLeastOne": 0.657215, "displayAtLeastOne": "65.7%", "pAtLeastTwo": 0.26321, "displayAtLeastTwo": "26.3%", "expectedPrizes": 1, "expectedFraction": "1/1", "pExactly": [ 0.342785, 0.394005, 0.196251, 0.055716, 0.009972, 0.001174, 0.000092, 0.000005 ] } ], "chart": [ { "yourTickets": 0, "pAtLeastOne": 0, "pAtLeastTwo": 0, "expectedPrizes": 0 }, { "yourTickets": 1, "pAtLeastOne": 0.033333, "pAtLeastTwo": 0, "expectedPrizes": 0.033333 }, { "yourTickets": 2, "pAtLeastOne": 0.065663, "pAtLeastTwo": 0.001003, "expectedPrizes": 0.066667 }, { "yourTickets": 3, "pAtLeastOne": 0.097017, "pAtLeastTwo": 0.002956, "expectedPrizes": 0.1 }, { "yourTickets": 4, "pAtLeastOne": 0.12742, "pAtLeastTwo": 0.005806, "expectedPrizes": 0.133333 }, { "yourTickets": 5, "pAtLeastOne": 0.156899, "pAtLeastTwo": 0.009504, "expectedPrizes": 0.166667 }, { "yourTickets": 6, "pAtLeastOne": 0.185479, "pAtLeastTwo": 0.014001, "expectedPrizes": 0.2 }, { "yourTickets": 7, "pAtLeastOne": 0.213184, "pAtLeastTwo": 0.01925, "expectedPrizes": 0.233333 }, { "yourTickets": 8, "pAtLeastOne": 0.240038, "pAtLeastTwo": 0.025207, "expectedPrizes": 0.266667 }, { "yourTickets": 9, "pAtLeastOne": 0.266064, "pAtLeastTwo": 0.031829, "expectedPrizes": 0.3 }, { "yourTickets": 10, "pAtLeastOne": 0.291285, "pAtLeastTwo": 0.039073, "expectedPrizes": 0.333333 }, { "yourTickets": 11, "pAtLeastOne": 0.315723, "pAtLeastTwo": 0.0469, "expectedPrizes": 0.366667 }, { "yourTickets": 12, "pAtLeastOne": 0.339401, "pAtLeastTwo": 0.055272, "expectedPrizes": 0.4 }, { "yourTickets": 13, "pAtLeastOne": 0.362338, "pAtLeastTwo": 0.064151, "expectedPrizes": 0.433333 }, { "yourTickets": 14, "pAtLeastOne": 0.384556, "pAtLeastTwo": 0.073502, "expectedPrizes": 0.466667 }, { "yourTickets": 15, "pAtLeastOne": 0.406075, "pAtLeastTwo": 0.08329, "expectedPrizes": 0.5 }, { "yourTickets": 16, "pAtLeastOne": 0.426915, "pAtLeastTwo": 0.093484, "expectedPrizes": 0.533333 }, { "yourTickets": 17, "pAtLeastOne": 0.447094, "pAtLeastTwo": 0.10405, "expectedPrizes": 0.566667 }, { "yourTickets": 18, "pAtLeastOne": 0.466631, "pAtLeastTwo": 0.11496, "expectedPrizes": 0.6 }, { "yourTickets": 19, "pAtLeastOne": 0.485545, "pAtLeastTwo": 0.126183, "expectedPrizes": 0.633333 }, { "yourTickets": 20, "pAtLeastOne": 0.503853, "pAtLeastTwo": 0.137693, "expectedPrizes": 0.666667 }, { "yourTickets": 21, "pAtLeastOne": 0.521573, "pAtLeastTwo": 0.149462, "expectedPrizes": 0.7 }, { "yourTickets": 22, "pAtLeastOne": 0.538721, "pAtLeastTwo": 0.161466, "expectedPrizes": 0.733333 }, { "yourTickets": 23, "pAtLeastOne": 0.555313, "pAtLeastTwo": 0.173679, "expectedPrizes": 0.766667 }, { "yourTickets": 24, "pAtLeastOne": 0.571367, "pAtLeastTwo": 0.186079, "expectedPrizes": 0.8 }, { "yourTickets": 25, "pAtLeastOne": 0.586897, "pAtLeastTwo": 0.198643, "expectedPrizes": 0.833333 }, { "yourTickets": 26, "pAtLeastOne": 0.601919, "pAtLeastTwo": 0.211349, "expectedPrizes": 0.866667 }, { "yourTickets": 27, "pAtLeastOne": 0.616448, "pAtLeastTwo": 0.224178, "expectedPrizes": 0.9 }, { "yourTickets": 28, "pAtLeastOne": 0.630497, "pAtLeastTwo": 0.23711, "expectedPrizes": 0.933333 }, { "yourTickets": 29, "pAtLeastOne": 0.644082, "pAtLeastTwo": 0.250127, "expectedPrizes": 0.966667 }, { "yourTickets": 30, "pAtLeastOne": 0.657215, "pAtLeastTwo": 0.26321, "expectedPrizes": 1 }, { "yourTickets": 31, "pAtLeastOne": 0.669911, "pAtLeastTwo": 0.276343, "expectedPrizes": 1.033333 }, { "yourTickets": 32, "pAtLeastOne": 0.682182, "pAtLeastTwo": 0.289511, "expectedPrizes": 1.066667 }, { "yourTickets": 33, "pAtLeastOne": 0.694041, "pAtLeastTwo": 0.302698, "expectedPrizes": 1.1 }, { "yourTickets": 34, "pAtLeastOne": 0.7055, "pAtLeastTwo": 0.315889, "expectedPrizes": 1.133333 }, { "yourTickets": 35, "pAtLeastOne": 0.716571, "pAtLeastTwo": 0.329072, "expectedPrizes": 1.166667 }, { "yourTickets": 36, "pAtLeastOne": 0.727267, "pAtLeastTwo": 0.342232, "expectedPrizes": 1.2 }, { "yourTickets": 37, "pAtLeastOne": 0.737598, "pAtLeastTwo": 0.355358, "expectedPrizes": 1.233333 }, { "yourTickets": 38, "pAtLeastOne": 0.747575, "pAtLeastTwo": 0.368439, "expectedPrizes": 1.266667 }, { "yourTickets": 39, "pAtLeastOne": 0.757209, "pAtLeastTwo": 0.381462, "expectedPrizes": 1.3 }, { "yourTickets": 40, "pAtLeastOne": 0.766512, "pAtLeastTwo": 0.394419, "expectedPrizes": 1.333333 }, { "yourTickets": 41, "pAtLeastOne": 0.775492, "pAtLeastTwo": 0.407299, "expectedPrizes": 1.366667 }, { "yourTickets": 42, "pAtLeastOne": 0.78416, "pAtLeastTwo": 0.420094, "expectedPrizes": 1.4 }, { "yourTickets": 43, "pAtLeastOne": 0.792526, "pAtLeastTwo": 0.432794, "expectedPrizes": 1.433333 }, { "yourTickets": 44, "pAtLeastOne": 0.800599, "pAtLeastTwo": 0.445391, "expectedPrizes": 1.466667 }, { "yourTickets": 45, "pAtLeastOne": 0.808388, "pAtLeastTwo": 0.457879, "expectedPrizes": 1.5 }, { "yourTickets": 46, "pAtLeastOne": 0.815902, "pAtLeastTwo": 0.47025, "expectedPrizes": 1.533333 }, { "yourTickets": 47, "pAtLeastOne": 0.82315, "pAtLeastTwo": 0.482497, "expectedPrizes": 1.566667 }, { "yourTickets": 48, "pAtLeastOne": 0.83014, "pAtLeastTwo": 0.494616, "expectedPrizes": 1.6 }, { "yourTickets": 49, "pAtLeastOne": 0.836881, "pAtLeastTwo": 0.506599, "expectedPrizes": 1.633333 }, { "yourTickets": 50, "pAtLeastOne": 0.84338, "pAtLeastTwo": 0.518441, "expectedPrizes": 1.666667 }, { "yourTickets": 51, "pAtLeastOne": 0.849645, "pAtLeastTwo": 0.530139, "expectedPrizes": 1.7 }, { "yourTickets": 52, "pAtLeastOne": 0.855683, "pAtLeastTwo": 0.541688, "expectedPrizes": 1.733333 }, { "yourTickets": 53, "pAtLeastOne": 0.861502, "pAtLeastTwo": 0.553083, "expectedPrizes": 1.766667 }, { "yourTickets": 54, "pAtLeastOne": 0.867109, "pAtLeastTwo": 0.56432, "expectedPrizes": 1.8 }, { "yourTickets": 55, "pAtLeastOne": 0.872511, "pAtLeastTwo": 0.575398, "expectedPrizes": 1.833333 }, { "yourTickets": 56, "pAtLeastOne": 0.877715, "pAtLeastTwo": 0.586312, "expectedPrizes": 1.866667 }, { "yourTickets": 57, "pAtLeastOne": 0.882727, "pAtLeastTwo": 0.597061, "expectedPrizes": 1.9 }, { "yourTickets": 58, "pAtLeastOne": 0.887553, "pAtLeastTwo": 0.607641, "expectedPrizes": 1.933333 }, { "yourTickets": 59, "pAtLeastOne": 0.892199, "pAtLeastTwo": 0.618051, "expectedPrizes": 1.966667 }, { "yourTickets": 60, "pAtLeastOne": 0.896672, "pAtLeastTwo": 0.628289, "expectedPrizes": 2 } ], "nextTicket": [ { "yourTickets": 0, "pAtLeastOne": 0, "addsToAtLeastOneFixedPool": 0.0333333, "addsToAtLeastOnePoolGrows": 0.0332226, "addsToExpectedFixedPool": 0.0333333, "addsToExpectedPoolGrows": 0.0332226 }, { "yourTickets": 1, "pAtLeastOne": 0.033333, "addsToAtLeastOneFixedPool": 0.03233, "addsToAtLeastOnePoolGrows": 0.0321152, "addsToExpectedFixedPool": 0.0333333, "addsToExpectedPoolGrows": 0.0331118 }, { "yourTickets": 2, "pAtLeastOne": 0.065663, "addsToAtLeastOneFixedPool": 0.0313536, "addsToAtLeastOnePoolGrows": 0.0310411, "addsToExpectedFixedPool": 0.0333333, "addsToExpectedPoolGrows": 0.0330011 }, { "yourTickets": 5, "pAtLeastOne": 0.156899, "addsToAtLeastOneFixedPool": 0.0285797, "addsToAtLeastOnePoolGrows": 0.02801, "addsToExpectedFixedPool": 0.0333333, "addsToExpectedPoolGrows": 0.0326689 }, { "yourTickets": 10, "pAtLeastOne": 0.291285, "addsToAtLeastOneFixedPool": 0.0244384, "addsToAtLeastOnePoolGrows": 0.0235454, "addsToExpectedFixedPool": 0.0333333, "addsToExpectedPoolGrows": 0.0321152 }, { "yourTickets": 20, "pAtLeastOne": 0.503853, "addsToAtLeastOneFixedPool": 0.0177195, "addsToAtLeastOnePoolGrows": 0.0164833, "addsToExpectedFixedPool": 0.0333333, "addsToExpectedPoolGrows": 0.0310078 }, { "yourTickets": 30, "pAtLeastOne": 0.657215, "addsToAtLeastOneFixedPool": 0.0126957, "addsToAtLeastOnePoolGrows": 0.0113882, "addsToExpectedFixedPool": 0.0333333, "addsToExpectedPoolGrows": 0.0299003 } ], "doubling10to20": { "prizes": 10, "expected10": 0.333333, "expected20": 0.666667, "expectedRatio": 2, "pAtLeastOne10": 0.291285, "pAtLeastOne20": 0.503853, "pAtLeastOneRatio": 1.7298, "display10": "29.1%", "display20": "50.4%" } }, "onePerPerson": { "others": 99, "prizes": 5, "formulas": { "personByDraw": "P(win draw j) = [prod_{i=0}^{j-2} (n-i)/(n+w-i)] × w/(n+w-j+1), n = rivals with 1 entry each", "personAny": "1 - prod_{i=0}^{m-1} (n-i)/(n+w-i)", "ticketsAny": "1 - C(n, m)/C(n+w, m): the same number. Until your first win the two rules draw from the same wheel (each rival who wins takes exactly one entry off either way)", "ticketsPerDraw": "w/(n+w) for every draw j", "ratioDrawToNext": "P(win draw j+1)/P(win draw j) = (n-j+1)/(n+w-j) < 1 when w > 1, so a multi-entry person is most likely to win the first prize drawn" }, "note": "Computed with exact fractions. pAnySameUnderBothRules is checked as an exact equality of fractions. With the person rule you can win at most one prize, so expectedPrizes = pAny. firstOverLast = P(win the prize drawn 1st)/P(win the prize drawn 5th). vsOneEntry.pAnyRatio = your chance of any prize with w entries divided by the chance with 1 entry (w = 20 is exact math only).", "rows": [ { "entries": 1, "simulated": true, "person": { "byDraw": [ 0.01, 0.01, 0.01, 0.01, 0.01 ], "byDrawDisplay": [ "1.00%", "1.00%", "1.00%", "1.00%", "1.00%" ], "pAny": 0.05, "displayAny": "5.00%", "expectedPrizes": 0.05, "firstOverLast": 1 }, "tickets": { "perDraw": 0.01, "perDrawFraction": "1/100", "perDrawDisplay": "1.00%", "pAny": 0.05, "displayAny": "5.00%", "pTwoOrMore": 0, "displayTwoOrMore": "0%", "expectedPrizes": 0.05 }, "pAnySameUnderBothRules": true, "vsOneEntry": { "pAnyRatio": 1 } }, { "entries": 2, "simulated": true, "person": { "byDraw": [ 0.019802, 0.019604, 0.019406, 0.019208, 0.01901 ], "byDrawDisplay": [ "1.98%", "1.96%", "1.94%", "1.92%", "1.90%" ], "pAny": 0.09703, "displayAny": "9.70%", "expectedPrizes": 0.09703, "firstOverLast": 1.0417 }, "tickets": { "perDraw": 0.019802, "perDrawFraction": "2/101", "perDrawDisplay": "1.98%", "pAny": 0.09703, "displayAny": "9.70%", "pTwoOrMore": 0.00198, "displayTwoOrMore": "0.20%", "expectedPrizes": 0.09901 }, "pAnySameUnderBothRules": true, "vsOneEntry": { "pAnyRatio": 1.9406 } }, { "entries": 3, "simulated": true, "person": { "byDraw": [ 0.029412, 0.028829, 0.028253, 0.027682, 0.027117 ], "byDrawDisplay": [ "2.94%", "2.88%", "2.83%", "2.77%", "2.71%" ], "pAny": 0.141293, "displayAny": "14.1%", "expectedPrizes": 0.141293, "firstOverLast": 1.0846 }, "tickets": { "perDraw": 0.029412, "perDrawFraction": "3/102", "perDrawDisplay": "2.94%", "pAny": 0.141293, "displayAny": "14.1%", "pTwoOrMore": 0.005708, "displayTwoOrMore": "0.57%", "expectedPrizes": 0.147059 }, "pAnySameUnderBothRules": true, "vsOneEntry": { "pAnyRatio": 2.8259 } }, { "entries": 5, "simulated": true, "person": { "byDraw": [ 0.048077, 0.04621, 0.044398, 0.042639, 0.040934 ], "byDrawDisplay": [ "4.81%", "4.62%", "4.44%", "4.26%", "4.09%" ], "pAny": 0.222258, "displayAny": "22.2%", "expectedPrizes": 0.222258, "firstOverLast": 1.1745 }, "tickets": { "perDraw": 0.048077, "perDrawFraction": "5/104", "perDrawDisplay": "4.81%", "pAny": 0.222258, "displayAny": "22.2%", "pTwoOrMore": 0.017589, "displayTwoOrMore": "1.76%", "expectedPrizes": 0.240385 }, "pAnySameUnderBothRules": true, "vsOneEntry": { "pAnyRatio": 4.4452 } }, { "entries": 10, "simulated": true, "person": { "byDraw": [ 0.091743, 0.084098, 0.077024, 0.070484, 0.064443 ], "byDrawDisplay": [ "9.17%", "8.41%", "7.70%", "7.05%", "6.44%" ], "pAny": 0.387792, "displayAny": "38.8%", "expectedPrizes": 0.387792, "firstOverLast": 1.4236 }, "tickets": { "perDraw": 0.091743, "perDrawFraction": "10/109", "perDrawDisplay": "9.17%", "pAny": 0.387792, "displayAny": "38.8%", "pTwoOrMore": 0.065578, "displayTwoOrMore": "6.56%", "expectedPrizes": 0.458716 }, "pAnySameUnderBothRules": true, "vsOneEntry": { "pAnyRatio": 7.7558 } }, { "entries": 20, "simulated": false, "person": { "byDraw": [ 0.168067, 0.141006, 0.118107, 0.098762, 0.082445 ], "byDrawDisplay": [ "16.8%", "14.1%", "11.8%", "9.88%", "8.24%" ], "pAny": 0.608387, "displayAny": "60.8%", "expectedPrizes": 0.608387, "firstOverLast": 2.0385 }, "tickets": { "perDraw": 0.168067, "perDrawFraction": "20/119", "perDrawDisplay": "16.8%", "pAny": 0.608387, "displayAny": "60.8%", "pTwoOrMore": 0.196163, "displayTwoOrMore": "19.6%", "expectedPrizes": 0.840336 }, "pAnySameUnderBothRules": true, "vsOneEntry": { "pAnyRatio": 12.1677 } } ] }, "mix": { "entries": 230, "people": 100, "prizes": 5, "method": "person rule: exact recursion over how many people of each group have won (exact fractions); ticket rule: hypergeometric (each line is a ticket, one line retired per win), every draw's winner is a uniformly random remaining line, so a group's share of every draw equals its share of the lines", "note": "Under the person rule a winner's whole weight leaves, so later draws lean toward 1-entry people: shareOfDraw falls draw by draw for the 10-entry group and rises for the 1-entry group. Under the ticket rule each group's share is the same for every draw, whatever the order. pAny differs between the rules here (it is identical only when every rival has one entry).", "groups": [ { "entries": 1, "people": 70, "person": { "shareOfDraw": [ 0.304348, 0.310487, 0.316797, 0.32328, 0.329937 ], "shareOfDrawDisplay": [ "30.4%", "31.0%", "31.7%", "32.3%", "33.0%" ], "perPersonByDraw": [ 0.004348, 0.004436, 0.004526, 0.004618, 0.004713 ], "perPersonByDrawDisplay": [ "0.43%", "0.44%", "0.45%", "0.46%", "0.47%" ], "perPersonAny": 0.022641, "perPersonAnyDisplay": "2.26%", "groupMeanPrizes": 1.584849 }, "tickets": { "shareOfEveryDraw": 0.304348, "shareOfEveryDrawFraction": "7/23", "shareDisplay": "30.4%", "perPersonPerDraw": 0.004348, "perPersonPerDrawFraction": "1/230", "perPersonPerDrawDisplay": "0.43%", "perPersonAny": 0.021739, "perPersonAnyDisplay": "2.17%", "perPersonTwoOrMore": 0, "perPersonTwoOrMoreDisplay": "0%", "perPersonExpectedPrizes": 0.021739, "groupMeanPrizes": 1.521739 } }, { "entries": 3, "people": 20, "person": { "shareOfDraw": [ 0.26087, 0.263824, 0.26679, 0.269764, 0.272739 ], "shareOfDrawDisplay": [ "26.1%", "26.4%", "26.7%", "27.0%", "27.3%" ], "perPersonByDraw": [ 0.013043, 0.013191, 0.013339, 0.013488, 0.013637 ], "perPersonByDrawDisplay": [ "1.30%", "1.32%", "1.33%", "1.35%", "1.36%" ], "perPersonAny": 0.066699, "perPersonAnyDisplay": "6.67%", "groupMeanPrizes": 1.333986 }, "tickets": { "shareOfEveryDraw": 0.26087, "shareOfEveryDrawFraction": "6/23", "shareDisplay": "26.1%", "perPersonPerDraw": 0.013043, "perPersonPerDrawFraction": "3/230", "perPersonPerDrawDisplay": "1.30%", "perPersonAny": 0.064083, "perPersonAnyDisplay": "6.41%", "perPersonTwoOrMore": 0.001129, "perPersonTwoOrMoreDisplay": "0.11%", "perPersonExpectedPrizes": 0.065217, "groupMeanPrizes": 1.304348 } }, { "entries": 10, "people": 10, "person": { "shareOfDraw": [ 0.434783, 0.425689, 0.416413, 0.406957, 0.397324 ], "shareOfDrawDisplay": [ "43.5%", "42.6%", "41.6%", "40.7%", "39.7%" ], "perPersonByDraw": [ 0.043478, 0.042569, 0.041641, 0.040696, 0.039732 ], "perPersonByDrawDisplay": [ "4.35%", "4.26%", "4.16%", "4.07%", "3.97%" ], "perPersonAny": 0.208117, "perPersonAnyDisplay": "20.8%", "groupMeanPrizes": 2.081165 }, "tickets": { "shareOfEveryDraw": 0.434783, "shareOfEveryDrawFraction": "10/23", "shareDisplay": "43.5%", "perPersonPerDraw": 0.043478, "perPersonPerDrawFraction": "1/23", "perPersonPerDrawDisplay": "4.35%", "perPersonAny": 0.200894, "perPersonAnyDisplay": "20.1%", "perPersonTwoOrMore": 0.015916, "perPersonTwoOrMoreDisplay": "1.59%", "perPersonExpectedPrizes": 0.217391, "groupMeanPrizes": 2.173913 } } ], "drawOrder": { "tenEntryPersonGrandPrizeFirst": 0.043478, "tenEntryPersonGrandPrizeLast": 0.039732, "tenEntryPersonDisplayFirst": "4.35%", "tenEntryPersonDisplayLast": "3.97%", "oneEntryPersonGrandPrizeFirst": 0.004348, "oneEntryPersonGrandPrizeLast": 0.004713, "oneEntryPersonDisplayFirst": "0.43%", "oneEntryPersonDisplayLast": "0.47%", "oneEntryGroupShareFirst": 0.304348, "oneEntryGroupShareLast": 0.329937, "oneEntryGroupShareFirstDisplay": "30.4%", "oneEntryGroupShareLastDisplay": "33.0%" } }, "bonus": { "people": 200, "baseEntries": 1, "bonusEntries": 4, "entriesPerSharer": 5, "prizes": 3, "formulas": { "onePrizeSharer": "5/(5S + 1(200-S)) with S sharers", "onePrizeNonSharer": "1/(5S + 1(200-S))", "sharersShareOfOdds": "5S/(5S + 1(200-S))", "threePrizes": "one prize per person, exact recursion over the draw sequence (exact fractions)" }, "note": "sharers = how many of the 200 entrants earned the bonus (S = 1 means you are the only one). Every entrant's chance depends on how many others took the bonus: when everyone takes it (S = 200) everyone is back to 1 in 200 for one prize. In the table nonSharer is null at S = 200. chart is from one entrant's side: othersSharing = how many of the other 199 took the bonus (0, 5, 10, ..., 195, then 199; the last step is 4), 'IfYouShare' = your chance if you take it too (othersSharing + 1 sharers), 'IfYouDont' = your chance if you do not (othersSharing sharers); no value is null. sharerOverNonSharer is below 5 with 3 prizes because a person can win only once.", "table": [ { "sharers": 1, "shareOfPeople": 0.005, "totalEntries": 204, "onePrize": { "sharer": 0.02451, "sharerDisplay": "2.45%", "sharerFraction": "5/204", "nonSharer": 0.004902, "nonSharerDisplay": "0.49%", "nonSharerFraction": "1/204", "sharersShareOfOdds": 0.02451, "sharersShareDisplay": "2.45%" }, "threePrizesOnePerPerson": { "sharerAny": 0.072088, "sharerDisplay": "7.21%", "nonSharerAny": 0.014713, "nonSharerDisplay": "1.47%", "sharerOverNonSharer": 4.8996 } }, { "sharers": 20, "shareOfPeople": 0.1, "totalEntries": 280, "onePrize": { "sharer": 0.017857, "sharerDisplay": "1.79%", "sharerFraction": "1/56", "nonSharer": 0.003571, "nonSharerDisplay": "0.36%", "nonSharerFraction": "1/280", "sharersShareOfOdds": 0.357143, "sharersShareDisplay": "35.7%" }, "threePrizesOnePerPerson": { "sharerAny": 0.053068, "sharerDisplay": "5.31%", "nonSharerAny": 0.01077, "nonSharerDisplay": "1.08%", "sharerOverNonSharer": 4.9273 } }, { "sharers": 50, "shareOfPeople": 0.25, "totalEntries": 400, "onePrize": { "sharer": 0.0125, "sharerDisplay": "1.25%", "sharerFraction": "1/80", "nonSharer": 0.0025, "nonSharerDisplay": "0.25%", "nonSharerFraction": "1/400", "sharersShareOfOdds": 0.625, "sharersShareDisplay": "62.5%" }, "threePrizesOnePerPerson": { "sharerAny": 0.037357, "sharerDisplay": "3.74%", "nonSharerAny": 0.007548, "nonSharerDisplay": "0.75%", "sharerOverNonSharer": 4.9494 } }, { "sharers": 100, "shareOfPeople": 0.5, "totalEntries": 600, "onePrize": { "sharer": 0.008333, "sharerDisplay": "0.83%", "sharerFraction": "1/120", "nonSharer": 0.001667, "nonSharerDisplay": "0.17%", "nonSharerFraction": "1/600", "sharersShareOfOdds": 0.833333, "sharersShareDisplay": "83.3%" }, "threePrizesOnePerPerson": { "sharerAny": 0.024972, "sharerDisplay": "2.50%", "nonSharerAny": 0.005028, "nonSharerDisplay": "0.50%", "sharerOverNonSharer": 4.9664 } }, { "sharers": 150, "shareOfPeople": 0.75, "totalEntries": 800, "onePrize": { "sharer": 0.00625, "sharerDisplay": "0.63%", "sharerFraction": "1/160", "nonSharer": 0.00125, "nonSharerDisplay": "0.13%", "nonSharerFraction": "1/800", "sharersShareOfOdds": 0.9375, "sharersShareDisplay": "93.75%" }, "threePrizesOnePerPerson": { "sharerAny": 0.018744, "sharerDisplay": "1.87%", "nonSharerAny": 0.003768, "nonSharerDisplay": "0.38%", "sharerOverNonSharer": 4.9748 } }, { "sharers": 200, "shareOfPeople": 1, "totalEntries": 1000, "onePrize": { "sharer": 0.005, "sharerDisplay": "0.50%", "sharerFraction": "1/200", "nonSharer": null, "nonSharerDisplay": null, "nonSharerFraction": null, "sharersShareOfOdds": 1, "sharersShareDisplay": "100%" }, "threePrizesOnePerPerson": { "sharerAny": 0.015, "sharerDisplay": "1.50%", "nonSharerAny": null, "nonSharerDisplay": null, "sharerOverNonSharer": null } } ], "chart": [ { "othersSharing": 0, "onePrizeIfYouShare": 0.02451, "onePrizeIfYouDont": 0.005, "threePrizesIfYouShare": 0.072088, "threePrizesIfYouDont": 0.015 }, { "othersSharing": 5, "onePrizeIfYouShare": 0.022321, "onePrizeIfYouDont": 0.004545, "threePrizesIfYouShare": 0.065903, "threePrizesIfYouDont": 0.013665 }, { "othersSharing": 10, "onePrizeIfYouShare": 0.020492, "onePrizeIfYouDont": 0.004167, "threePrizesIfYouShare": 0.060676, "threePrizesIfYouDont": 0.012544 }, { "othersSharing": 15, "onePrizeIfYouShare": 0.018939, "onePrizeIfYouDont": 0.003846, "threePrizesIfYouShare": 0.056204, "threePrizesIfYouDont": 0.011591 }, { "othersSharing": 20, "onePrizeIfYouShare": 0.017606, "onePrizeIfYouDont": 0.003571, "threePrizesIfYouShare": 0.052337, "threePrizesIfYouDont": 0.01077 }, { "othersSharing": 25, "onePrizeIfYouShare": 0.016447, "onePrizeIfYouDont": 0.003333, "threePrizesIfYouShare": 0.048962, "threePrizesIfYouDont": 0.010057 }, { "othersSharing": 30, "onePrizeIfYouShare": 0.015432, "onePrizeIfYouDont": 0.003125, "threePrizesIfYouShare": 0.045992, "threePrizesIfYouDont": 0.009431 }, { "othersSharing": 35, "onePrizeIfYouShare": 0.014535, "onePrizeIfYouDont": 0.002941, "threePrizesIfYouShare": 0.043358, "threePrizesIfYouDont": 0.008878 }, { "othersSharing": 40, "onePrizeIfYouShare": 0.013736, "onePrizeIfYouDont": 0.002778, "threePrizesIfYouShare": 0.041007, "threePrizesIfYouDont": 0.008386 }, { "othersSharing": 45, "onePrizeIfYouShare": 0.013021, "onePrizeIfYouDont": 0.002632, "threePrizesIfYouShare": 0.038896, "threePrizesIfYouDont": 0.007945 }, { "othersSharing": 50, "onePrizeIfYouShare": 0.012376, "onePrizeIfYouDont": 0.0025, "threePrizesIfYouShare": 0.036991, "threePrizesIfYouDont": 0.007548 }, { "othersSharing": 55, "onePrizeIfYouShare": 0.011792, "onePrizeIfYouDont": 0.002381, "threePrizesIfYouShare": 0.035262, "threePrizesIfYouDont": 0.007188 }, { "othersSharing": 60, "onePrizeIfYouShare": 0.011261, "onePrizeIfYouDont": 0.002273, "threePrizesIfYouShare": 0.033687, "threePrizesIfYouDont": 0.006861 }, { "othersSharing": 65, "onePrizeIfYouShare": 0.010776, "onePrizeIfYouDont": 0.002174, "threePrizesIfYouShare": 0.032246, "threePrizesIfYouDont": 0.006562 }, { "othersSharing": 70, "onePrizeIfYouShare": 0.010331, "onePrizeIfYouDont": 0.002083, "threePrizesIfYouShare": 0.030922, "threePrizesIfYouDont": 0.006289 }, { "othersSharing": 75, "onePrizeIfYouShare": 0.009921, "onePrizeIfYouDont": 0.002, "threePrizesIfYouShare": 0.029703, "threePrizesIfYouDont": 0.006037 }, { "othersSharing": 80, "onePrizeIfYouShare": 0.009542, "onePrizeIfYouDont": 0.001923, "threePrizesIfYouShare": 0.028576, "threePrizesIfYouDont": 0.005804 }, { "othersSharing": 85, "onePrizeIfYouShare": 0.009191, "onePrizeIfYouDont": 0.001852, "threePrizesIfYouShare": 0.02753, "threePrizesIfYouDont": 0.005588 }, { "othersSharing": 90, "onePrizeIfYouShare": 0.008865, "onePrizeIfYouDont": 0.001786, "threePrizesIfYouShare": 0.026559, "threePrizesIfYouDont": 0.005388 }, { "othersSharing": 95, "onePrizeIfYouShare": 0.008562, "onePrizeIfYouDont": 0.001724, "threePrizesIfYouShare": 0.025653, "threePrizesIfYouDont": 0.005202 }, { "othersSharing": 100, "onePrizeIfYouShare": 0.008278, "onePrizeIfYouDont": 0.001667, "threePrizesIfYouShare": 0.024807, "threePrizesIfYouDont": 0.005028 }, { "othersSharing": 105, "onePrizeIfYouShare": 0.008013, "onePrizeIfYouDont": 0.001613, "threePrizesIfYouShare": 0.024015, "threePrizesIfYouDont": 0.004866 }, { "othersSharing": 110, "onePrizeIfYouShare": 0.007764, "onePrizeIfYouDont": 0.001563, "threePrizesIfYouShare": 0.023272, "threePrizesIfYouDont": 0.004713 }, { "othersSharing": 115, "onePrizeIfYouShare": 0.00753, "onePrizeIfYouDont": 0.001515, "threePrizesIfYouShare": 0.022573, "threePrizesIfYouDont": 0.00457 }, { "othersSharing": 120, "onePrizeIfYouShare": 0.00731, "onePrizeIfYouDont": 0.001471, "threePrizesIfYouShare": 0.021915, "threePrizesIfYouDont": 0.004435 }, { "othersSharing": 125, "onePrizeIfYouShare": 0.007102, "onePrizeIfYouDont": 0.001429, "threePrizesIfYouShare": 0.021294, "threePrizesIfYouDont": 0.004308 }, { "othersSharing": 130, "onePrizeIfYouShare": 0.006906, "onePrizeIfYouDont": 0.001389, "threePrizesIfYouShare": 0.020707, "threePrizesIfYouDont": 0.004188 }, { "othersSharing": 135, "onePrizeIfYouShare": 0.00672, "onePrizeIfYouDont": 0.001351, "threePrizesIfYouShare": 0.020152, "threePrizesIfYouDont": 0.004074 }, { "othersSharing": 140, "onePrizeIfYouShare": 0.006545, "onePrizeIfYouDont": 0.001316, "threePrizesIfYouShare": 0.019625, "threePrizesIfYouDont": 0.003967 }, { "othersSharing": 145, "onePrizeIfYouShare": 0.006378, "onePrizeIfYouDont": 0.001282, "threePrizesIfYouShare": 0.019126, "threePrizesIfYouDont": 0.003865 }, { "othersSharing": 150, "onePrizeIfYouShare": 0.006219, "onePrizeIfYouDont": 0.00125, "threePrizesIfYouShare": 0.018651, "threePrizesIfYouDont": 0.003768 }, { "othersSharing": 155, "onePrizeIfYouShare": 0.006068, "onePrizeIfYouDont": 0.00122, "threePrizesIfYouShare": 0.018199, "threePrizesIfYouDont": 0.003676 }, { "othersSharing": 160, "onePrizeIfYouShare": 0.005924, "onePrizeIfYouDont": 0.00119, "threePrizesIfYouShare": 0.017769, "threePrizesIfYouDont": 0.003588 }, { "othersSharing": 165, "onePrizeIfYouShare": 0.005787, "onePrizeIfYouDont": 0.001163, "threePrizesIfYouShare": 0.017358, "threePrizesIfYouDont": 0.003504 }, { "othersSharing": 170, "onePrizeIfYouShare": 0.005656, "onePrizeIfYouDont": 0.001136, "threePrizesIfYouShare": 0.016966, "threePrizesIfYouDont": 0.003424 }, { "othersSharing": 175, "onePrizeIfYouShare": 0.005531, "onePrizeIfYouDont": 0.001111, "threePrizesIfYouShare": 0.016591, "threePrizesIfYouDont": 0.003348 }, { "othersSharing": 180, "onePrizeIfYouShare": 0.005411, "onePrizeIfYouDont": 0.001087, "threePrizesIfYouShare": 0.016232, "threePrizesIfYouDont": 0.003275 }, { "othersSharing": 185, "onePrizeIfYouShare": 0.005297, "onePrizeIfYouDont": 0.001064, "threePrizesIfYouShare": 0.015889, "threePrizesIfYouDont": 0.003205 }, { "othersSharing": 190, "onePrizeIfYouShare": 0.005187, "onePrizeIfYouDont": 0.001042, "threePrizesIfYouShare": 0.01556, "threePrizesIfYouDont": 0.003138 }, { "othersSharing": 195, "onePrizeIfYouShare": 0.005081, "onePrizeIfYouDont": 0.00102, "threePrizesIfYouShare": 0.015244, "threePrizesIfYouDont": 0.003074 }, { "othersSharing": 199, "onePrizeIfYouShare": 0.005, "onePrizeIfYouDont": 0.001004, "threePrizesIfYouShare": 0.015, "threePrizesIfYouDont": 0.003024 } ] } }, "checks": [ { "name": "big book, second prize, weighted blocks: the first winner's block wins again", "simulated": 0.098448, "exact": 0.099099, "se": 0.000423, "z": -1.54, "pass": true }, { "name": "big book, second prize, weighted blocks: every remaining ticket equally likely (chi-square, 998 df)", "simulated": 1031.738728, "exact": 998, "se": 44.7, "z": 0.76, "pass": true }, { "name": "one prize, 100 tickets: every ticket equally likely (chi-square, 99 df)", "simulated": 93.5468, "exact": 99, "se": 14.1, "z": -0.39, "pass": true }, { "name": "one prize: hold 1 of 100 tickets", "simulated": 0.010102, "exact": 0.01, "se": 0.0000995, "z": 1.03, "pass": true }, { "name": "one prize: hold 5 of 100 tickets", "simulated": 0.050137, "exact": 0.05, "se": 0.000218, "z": 0.63, "pass": true }, { "name": "one prize: hold 10 of 100 tickets", "simulated": 0.100357, "exact": 0.1, "se": 0.0003, "z": 1.19, "pass": true }, { "name": "one prize: hold 20 of 100 tickets", "simulated": 0.20006, "exact": 0.2, "se": 0.0004, "z": 0.15, "pass": true }, { "name": "one prize, 101 tickets: every ticket equally likely (chi-square, 100 df)", "simulated": 102.907586, "exact": 100, "se": 14.1, "z": 0.21, "pass": true }, { "name": "one prize after buying one more: hold 2 of 101 tickets", "simulated": 0.019667, "exact": 0.019802, "se": 0.000139, "z": -0.97, "pass": true }, { "name": "one prize after buying one more: hold 6 of 101 tickets", "simulated": 0.059272, "exact": 0.059406, "se": 0.000236, "z": -0.57, "pass": true }, { "name": "one prize after buying one more: hold 11 of 101 tickets", "simulated": 0.108541, "exact": 0.108911, "se": 0.000312, "z": -1.19, "pass": true }, { "name": "one prize, 300 tickets: every ticket equally likely (chi-square, 299 df)", "simulated": 299.9936, "exact": 299, "se": 24.5, "z": 0.04, "pass": true }, { "name": "one prize: hold 1 of 300 tickets", "simulated": 0.003332, "exact": 0.003333, "se": 0.0000576, "z": -0.02, "pass": true }, { "name": "one prize: hold 5 of 300 tickets", "simulated": 0.016788, "exact": 0.016667, "se": 0.000128, "z": 0.95, "pass": true }, { "name": "one prize: hold 10 of 300 tickets", "simulated": 0.033295, "exact": 0.033333, "se": 0.00018, "z": -0.21, "pass": true }, { "name": "one prize: hold 20 of 300 tickets", "simulated": 0.066264, "exact": 0.066667, "se": 0.000249, "z": -1.61, "pass": true }, { "name": "one prize, 301 tickets: every ticket equally likely (chi-square, 300 df)", "simulated": 322.059702, "exact": 300, "se": 24.5, "z": 0.9, "pass": true }, { "name": "one prize after buying one more: hold 2 of 301 tickets", "simulated": 0.006633, "exact": 0.006645, "se": 0.0000812, "z": -0.14, "pass": true }, { "name": "one prize after buying one more: hold 6 of 301 tickets", "simulated": 0.019773, "exact": 0.019934, "se": 0.00014, "z": -1.15, "pass": true }, { "name": "one prize after buying one more: hold 11 of 301 tickets", "simulated": 0.03633, "exact": 0.036545, "se": 0.000188, "z": -1.15, "pass": true }, { "name": "one prize, 1000 tickets: every ticket equally likely (chi-square, 999 df)", "simulated": 1003.47, "exact": 999, "se": 44.7, "z": 0.1, "pass": true }, { "name": "one prize: hold 1 of 1000 tickets", "simulated": 0.00102, "exact": 0.001, "se": 0.0000316, "z": 0.63, "pass": true }, { "name": "one prize: hold 5 of 1000 tickets", "simulated": 0.005045, "exact": 0.005, "se": 0.0000705, "z": 0.64, "pass": true }, { "name": "one prize: hold 10 of 1000 tickets", "simulated": 0.01006, "exact": 0.01, "se": 0.0000995, "z": 0.6, "pass": true }, { "name": "one prize: hold 20 of 1000 tickets", "simulated": 0.020234, "exact": 0.02, "se": 0.00014, "z": 1.67, "pass": true }, { "name": "3 prizes, tickets retired: P(at least one) holding 1 of 300", "simulated": 0.010208, "exact": 0.01, "se": 0.000141, "z": 1.48, "pass": true }, { "name": "3 prizes, tickets retired: P(at least one) holding 5 of 300", "simulated": 0.04965, "exact": 0.049333, "se": 0.000306, "z": 1.03, "pass": true }, { "name": "3 prizes, tickets retired: P(two or more) holding 5 of 300", "simulated": 0.000706, "exact": 0.000664, "se": 0.0000364, "z": 1.14, "pass": true }, { "name": "3 prizes, tickets retired: P(at least one) holding 10 of 300", "simulated": 0.097282, "exact": 0.097017, "se": 0.000419, "z": 0.63, "pass": true }, { "name": "3 prizes, tickets retired: P(two or more) holding 10 of 300", "simulated": 0.002996, "exact": 0.002956, "se": 0.0000768, "z": 0.52, "pass": true }, { "name": "3 prizes, tickets retired: P(at least one) holding 20 of 300", "simulated": 0.18775, "exact": 0.187547, "se": 0.000552, "z": 0.37, "pass": true }, { "name": "3 prizes, tickets retired: P(two or more) holding 20 of 300", "simulated": 0.012206, "exact": 0.012197, "se": 0.000155, "z": 0.06, "pass": true }, { "name": "5 prizes, tickets retired: P(at least one) holding 1 of 300", "simulated": 0.016706, "exact": 0.016667, "se": 0.000181, "z": 0.22, "pass": true }, { "name": "5 prizes, tickets retired: P(at least one) holding 5 of 300", "simulated": 0.081378, "exact": 0.081126, "se": 0.000386, "z": 0.65, "pass": true }, { "name": "5 prizes, tickets retired: P(two or more) holding 5 of 300", "simulated": 0.002266, "exact": 0.002185, "se": 0.000066, "z": 1.23, "pass": true }, { "name": "5 prizes, tickets retired: P(at least one) holding 10 of 300", "simulated": 0.157294, "exact": 0.156899, "se": 0.000514, "z": 0.77, "pass": true }, { "name": "5 prizes, tickets retired: P(two or more) holding 10 of 300", "simulated": 0.00937, "exact": 0.009504, "se": 0.000137, "z": -0.98, "pass": true }, { "name": "5 prizes, tickets retired: P(at least one) holding 20 of 300", "simulated": 0.293738, "exact": 0.293456, "se": 0.000644, "z": 0.44, "pass": true }, { "name": "5 prizes, tickets retired: P(two or more) holding 20 of 300", "simulated": 0.03718, "exact": 0.037462, "se": 0.000269, "z": -1.05, "pass": true }, { "name": "10 prizes, tickets retired: P(at least one) holding 1 of 300", "simulated": 0.033482, "exact": 0.033333, "se": 0.000254, "z": 0.59, "pass": true }, { "name": "10 prizes, tickets retired: P(at least one) holding 5 of 300", "simulated": 0.156582, "exact": 0.156899, "se": 0.000514, "z": -0.62, "pass": true }, { "name": "10 prizes, tickets retired: P(two or more) holding 5 of 300", "simulated": 0.009644, "exact": 0.009504, "se": 0.000137, "z": 1.02, "pass": true }, { "name": "10 prizes, tickets retired: expected prizes holding 5 of 300", "simulated": 0.166476, "exact": 0.166667, "se": 0.000564, "z": -0.34, "pass": true }, { "name": "10 prizes, tickets retired: P(at least one) holding 10 of 300", "simulated": 0.291456, "exact": 0.291285, "se": 0.000643, "z": 0.27, "pass": true }, { "name": "10 prizes, tickets retired: P(two or more) holding 10 of 300", "simulated": 0.039086, "exact": 0.039073, "se": 0.000274, "z": 0.05, "pass": true }, { "name": "10 prizes, tickets retired: expected prizes holding 10 of 300", "simulated": 0.333536, "exact": 0.333333, "se": 0.000791, "z": 0.26, "pass": true }, { "name": "10 prizes, tickets retired: P(at least one) holding 20 of 300", "simulated": 0.50359, "exact": 0.503853, "se": 0.000707, "z": -0.37, "pass": true }, { "name": "10 prizes, tickets retired: P(two or more) holding 20 of 300", "simulated": 0.137964, "exact": 0.137693, "se": 0.000487, "z": 0.56, "pass": true }, { "name": "10 prizes, tickets retired: expected prizes holding 20 of 300", "simulated": 0.66659, "exact": 0.666667, "se": 0.0011, "z": -0.07, "pass": true }, { "name": "10 prizes, tickets retired: expected prizes holding 30 of 300", "simulated": 0.9992, "exact": 1, "se": 0.00132, "z": -0.61, "pass": true }, { "name": "tickets retired: holder of 10 of 300 wins the prize drawn 1st", "simulated": 0.033566, "exact": 0.033333, "se": 0.000254, "z": 0.92, "pass": true }, { "name": "tickets retired: holder of 10 of 300 wins the prize drawn 5th", "simulated": 0.033408, "exact": 0.033333, "se": 0.000254, "z": 0.29, "pass": true }, { "name": "tickets retired: holder of 10 of 300 wins the prize drawn 10th", "simulated": 0.03279, "exact": 0.033333, "se": 0.000254, "z": -2.14, "pass": true }, { "name": "tickets retired: every ticket equally likely to win the 10th-drawn prize (chi-square, 299 df)", "simulated": 300.0112, "exact": 299, "se": 24.5, "z": 0.04, "pass": true }, { "name": "one prize per person: 1 entry vs 99 people with 1, wins the prize drawn 5th", "simulated": 0.010088, "exact": 0.01, "se": 0.000128, "z": 0.69, "pass": true }, { "name": "one prize per person: 1 entry vs 99 people with 1, wins any of 5 prizes", "simulated": 0.05041, "exact": 0.05, "se": 0.000281, "z": 1.46, "pass": true }, { "name": "one prize per person: 2 entries vs 99 people with 1, wins the prize drawn 1st", "simulated": 0.020067, "exact": 0.019802, "se": 0.00018, "z": 1.47, "pass": true }, { "name": "one prize per person: 2 entries vs 99 people with 1, wins the prize drawn 5th", "simulated": 0.01921, "exact": 0.01901, "se": 0.000176, "z": 1.14, "pass": true }, { "name": "one prize per person: 2 entries vs 99 people with 1, wins any of 5 prizes", "simulated": 0.097628, "exact": 0.09703, "se": 0.000382, "z": 1.57, "pass": true }, { "name": "one prize per person: 3 entries vs 99 people with 1, wins the prize drawn 1st", "simulated": 0.029283, "exact": 0.029412, "se": 0.000218, "z": -0.59, "pass": true }, { "name": "one prize per person: 3 entries vs 99 people with 1, wins the prize drawn 5th", "simulated": 0.02677, "exact": 0.027117, "se": 0.00021, "z": -1.66, "pass": true }, { "name": "one prize per person: 3 entries vs 99 people with 1, wins any of 5 prizes", "simulated": 0.140567, "exact": 0.141293, "se": 0.00045, "z": -1.61, "pass": true }, { "name": "one prize per person: 5 entries vs 99 people with 1, wins the prize drawn 1st", "simulated": 0.048098, "exact": 0.048077, "se": 0.000276, "z": 0.08, "pass": true }, { "name": "one prize per person: 5 entries vs 99 people with 1, wins the prize drawn 5th", "simulated": 0.04081, "exact": 0.040934, "se": 0.000256, "z": -0.48, "pass": true }, { "name": "one prize per person: 5 entries vs 99 people with 1, wins any of 5 prizes", "simulated": 0.221918, "exact": 0.222258, "se": 0.000537, "z": -0.63, "pass": true }, { "name": "one prize per person: 10 entries vs 99 people with 1, wins the prize drawn 1st", "simulated": 0.091682, "exact": 0.091743, "se": 0.000373, "z": -0.16, "pass": true }, { "name": "one prize per person: 10 entries vs 99 people with 1, wins the prize drawn 5th", "simulated": 0.064697, "exact": 0.064443, "se": 0.000317, "z": 0.8, "pass": true }, { "name": "one prize per person: 10 entries vs 99 people with 1, wins any of 5 prizes", "simulated": 0.38761, "exact": 0.387792, "se": 0.000629, "z": -0.29, "pass": true }, { "name": "tickets retired: 5 lines vs 99 people with 1, wins the prize drawn 5th", "simulated": 0.048527, "exact": 0.048077, "se": 0.000276, "z": 1.63, "pass": true }, { "name": "tickets retired: 5 lines vs 99 people with 1, wins at least one of 5", "simulated": 0.222552, "exact": 0.222258, "se": 0.000537, "z": 0.55, "pass": true }, { "name": "tickets retired: 5 lines vs 99 people with 1, wins two or more of 5", "simulated": 0.017783, "exact": 0.017589, "se": 0.00017, "z": 1.15, "pass": true }, { "name": "tickets retired: 10 lines vs 99 people with 1, wins the prize drawn 5th", "simulated": 0.091922, "exact": 0.091743, "se": 0.000373, "z": 0.48, "pass": true }, { "name": "tickets retired: 10 lines vs 99 people with 1, wins at least one of 5", "simulated": 0.386172, "exact": 0.387792, "se": 0.000629, "z": -2.58, "pass": true }, { "name": "tickets retired: 10 lines vs 99 people with 1, wins two or more of 5", "simulated": 0.065217, "exact": 0.065578, "se": 0.00032, "z": -1.13, "pass": true }, { "name": "mixed giveaway, one prize per person: 1-entry people win the prize drawn 1st", "simulated": 0.303532, "exact": 0.304348, "se": 0.000594, "z": -1.37, "pass": true }, { "name": "mixed giveaway, one prize per person: 1-entry people win the prize drawn 5th", "simulated": 0.330668, "exact": 0.329937, "se": 0.000607, "z": 1.21, "pass": true }, { "name": "mixed giveaway, one prize per person: prizes won by the 1-entry group", "simulated": 1.584833, "exact": 1.584849, "se": 0.00131, "z": -0.01, "pass": true }, { "name": "mixed giveaway, tickets retired: 1-entry people win the prize drawn 5th", "simulated": 0.304823, "exact": 0.304348, "se": 0.000594, "z": 0.8, "pass": true }, { "name": "mixed giveaway, tickets retired: prizes won by the 1-entry group", "simulated": 1.523475, "exact": 1.521739, "se": 0.00132, "z": 1.32, "pass": true }, { "name": "mixed giveaway, one prize per person: 3-entry people win the prize drawn 1st", "simulated": 0.2618, "exact": 0.26087, "se": 0.000567, "z": 1.64, "pass": true }, { "name": "mixed giveaway, one prize per person: 3-entry people win the prize drawn 5th", "simulated": 0.272, "exact": 0.272739, "se": 0.000575, "z": -1.29, "pass": true }, { "name": "mixed giveaway, one prize per person: prizes won by the 3-entry group", "simulated": 1.3336, "exact": 1.333986, "se": 0.00123, "z": -0.31, "pass": true }, { "name": "mixed giveaway, tickets retired: 3-entry people win the prize drawn 5th", "simulated": 0.260283, "exact": 0.26087, "se": 0.000567, "z": -1.03, "pass": true }, { "name": "mixed giveaway, tickets retired: prizes won by the 3-entry group", "simulated": 1.304552, "exact": 1.304348, "se": 0.00126, "z": 0.16, "pass": true }, { "name": "mixed giveaway, tickets retired: one 3-entry person wins two or more (pooled over 20 people)", "simulated": 0.001128, "exact": 0.001129, "se": 0.00000961, "z": -0.12, "pass": true }, { "name": "mixed giveaway, tickets retired: one 3-entry person wins at least one (pooled over 20 people)", "simulated": 0.064095, "exact": 0.064083, "se": 0.0000614, "z": 0.19, "pass": true }, { "name": "mixed giveaway, one prize per person: 10-entry people win the prize drawn 1st", "simulated": 0.434668, "exact": 0.434783, "se": 0.00064, "z": -0.18, "pass": true }, { "name": "mixed giveaway, one prize per person: 10-entry people win the prize drawn 5th", "simulated": 0.397332, "exact": 0.397324, "se": 0.000632, "z": 0.01, "pass": true }, { "name": "mixed giveaway, one prize per person: prizes won by the 10-entry group", "simulated": 2.081567, "exact": 2.081165, "se": 0.00134, "z": 0.3, "pass": true }, { "name": "mixed giveaway, tickets retired: 10-entry people win the prize drawn 5th", "simulated": 0.434893, "exact": 0.434783, "se": 0.00064, "z": 0.17, "pass": true }, { "name": "mixed giveaway, tickets retired: prizes won by the 10-entry group", "simulated": 2.171973, "exact": 2.173913, "se": 0.00142, "z": -1.37, "pass": true }, { "name": "mixed giveaway, tickets retired: one 10-entry person wins two or more (pooled over 10 people)", "simulated": 0.015938, "exact": 0.015916, "se": 0.0000486, "z": 0.46, "pass": true }, { "name": "mixed giveaway, tickets retired: one 10-entry person wins at least one (pooled over 10 people)", "simulated": 0.200667, "exact": 0.200894, "se": 0.000129, "z": -1.76, "pass": true }, { "name": "bonus entries (50 of 200 share, 5 entries each): a sharer's group wins the 1st-drawn prize", "simulated": 0.623734, "exact": 0.625, "se": 0.000685, "z": -1.85, "pass": true }, { "name": "bonus entries (50 of 200 share): a sharer's group wins the 3rd-drawn prize", "simulated": 0.62042, "exact": 0.620214, "se": 0.000686, "z": 0.3, "pass": true }, { "name": "bonus entries (50 of 200 share): one sharer's chance of any of 3 prizes (group mean / 50)", "simulated": 0.037347, "exact": 0.037357, "se": 0.0000236, "z": -0.43, "pass": true }, { "name": "bonus entries (50 of 200 share): one particular non-sharer wins any of 3 prizes", "simulated": 0.007574, "exact": 0.007548, "se": 0.000122, "z": 0.21, "pass": true } ] }