{ "license": "CC BY 4.0 — https://creativecommons.org/licenses/by/4.0/", "attribution": "Wheel Cue (wheelcue.com)", "source": "https://wheelcue.com/research/is-a-coin-flip-really-50-50/", "slug": "is-a-coin-flip-really-50-50", "generatedAt": "2026-10-10T14:45:46.086Z", "node": "v24.21.0", "randomJsSha256": "946e82a3c426d698779d3246c37c732f771db3873c93239ef3952aea39be1e2d", "totalSpins": 307297285, "rawDraws": 307297297, "totalFlips": 305297285, "virtualFlips": 100000000, "handFlips": 205297285, "startDraws": 2000000, "seconds": 14.3, "checksTotal": 26, "checksPassed": 26, "params": { "coin": { "sameSide": 0.508, "sameSidePerMille": 508, "source": "Bartoš et al. (2025), share pooled over all 350,757 flips (not the average across people); Diaconis, Holmes & Montgomery (2007) quote about 0.51, which they say is the mean of their estimates, 0.508, rounded up (sources.dhm.meanEstimate), so it is the same figure, not a separate, larger prediction", "alsoExact": [ 0.505, 0.51, 0.515, 0.52, 0.53, 0.55, 0.6 ] }, "test": { "rule": "exact binomial test of 'more same-side landings than chance': flag the coin when the same-side count X >= c(n), where c(n) is the smallest count with P(X >= c | fair) <= alpha", "alpha": 0.05, "oneSided": true, "alsoTwoSided": true, "powers": [ 0.8, 0.9 ] }, "detection": { "sessions": { "100": 200000, "1000": 40000 }, "fairSessions": { "100": 200000, "1000": 40000 }, "powerSessionsAtSampleSize": 3000, "flipsTable": [ 10, 20, 50, 100, 200, 500, 1000, 2000, 5000, 10000, 20000, 50000, 100000, 200000, 350757, 500000, 1000000 ], "widgetFlips": [ 10, 20, 50, 100, 200, 500, 1000, 2000, 5000, 10000 ], "chart": { "from": 10, "to": 1000000, "pointsPerDecade": 20 } }, "study": { "flips": 350757, "replications": 100 }, "games": { "bestOf": [ 1, 3, 5, 7 ], "matches": 1000000, "callTosses": 2000000 }, "streaks": { "flips": 100, "sessions": 200000, "thresholds": [ 5, 6, 7, 8, 10 ], "distributionUpTo": 20 }, "virtual": { "flips": 20000000 } }, "notes": { "simulatedVsExact": "results holds simulated flips (never site usage); theory holds exact math (exact fractions, or exact binomial sums in double precision cross-checked against fractions) rounded to 6 decimals. Quote theory values; call results values 'simulated'. sources holds the two papers' own figures: attribute them to the papers, never call them ours.", "siteBehavior": "The yes or no wheel (/tools/yes-or-no-wheel/) starts with the entries Yes and No (remove-winner off); its presets are Yes / No, Add a maybe (Yes, No, Maybe), Do it / Don't, Now / Later. /templates/who-goes-first/ has a 'Heads or tails' preset (#heads-or-tails) with the entries Heads and Tails. Every spin is CueRandom.pickWeighted() over the entries' weights (app.js), drawn before the animation starts. With two equal entries that is below(2): exactly 1/2, no rejection ever, independent of every earlier spin. The site's coin has no starting side, so the same-side effect cannot arise.", "handFlipModel": "A hand flip is modeled as one below(1000) draw: lands on its starting side when the index is below 508, so the chance is exactly 508/1000 = 127/250. 0.508 is the share pooled over all 350,757 flips in Bartoš et al. (not the average across people; their hierarchical estimate for a typical flipper is 0.5098). Scope: every exact and simulated hand-flip value is for one coin (or flipper) with exactly a 50.8% same-side chance on every flip, independently. Real people differ (0.487 to 0.601 in Bartoš et al.'s Table 1) and the bias fell as they flipped more, so write 'a coin (or flipper) with a 50.8% same-side chance' and point to the widget for other values. The streak comparison also assumes every flip starts from the side it last landed on, as in the study. The model says nothing about coins that bounce or spin on a table.", "spins": "totalSpins counts every CueRandom call that picks a result: virtualFlips (pickWeighted([1,1]), one spin of a two-entry wheel), handFlips (below(1000) in the hand-flip model) and startDraws (below(2) for a hidden starting side). totalFlips = virtualFlips + handFlips. totalSpins includes startDraws (hidden starting sides), which are not coin flips, and changes slightly from run to run because best-of matches have variable length. In the article set drawsLabel: 'simulated random draws' for totalSpins, or quote totalFlips as simulated coin flips, always read from the JSON. rawDraws counts 32-bit numbers handed to random.js, including the few the rejection step of below(1000) discards.", "checks": "Each check: z = (simulated - exact)/se, pass when |z| <= 4. The pair check uses Good's serial statistic (overlapping-pair chi-square minus single-count chi-square), which is chi-square with 2 degrees of freedom for independent fair flips: exact = 2, se = sqrt(2 × 2) = 2. (The overlapping-pair chi-square alone is not chi-square with 3 df: neighboring pairs share a flip, and its variance is about 10, not 6.) The spread check compares the sample SD of 100 replications with the exact SD, se = SD/sqrt(2 × 99). The rejection sanity check must be exactly 0. Simulation-only values (lowestShare/highestShare of the replications, pair counts, the two chi-square parts) are not checked.", "rounding": "display: two decimals below 10% or above 90%, one between; display2 (games): always two decimals, because the differences are fractions of a point; 'points' are percentage points with two decimals." }, "sources": { "note": "Bibliographic details and the figures this study uses, as verified on the pages and full texts listed (October 2026). Quote at most one short sentence. Every value here is the papers', not ours.", "bartos": { "title": "Fair Coins Tend to Land on the Same Side They Started: Evidence from 350,757 Flips", "authors": "František Bartoš, Alexandra Sarafoglou, Henrik R. Godmann, Amir Sahrani, … Eric-Jan Wagenmakers (50 authors)", "shortAuthors": "Bartoš et al.", "journal": "Journal of the American Statistical Association", "volume": 120, "issue": 552, "pages": "2118–2127", "year": 2025, "doi": "10.1080/01621459.2025.2516210", "doiUrl": "https://doi.org/10.1080/01621459.2025.2516210", "preprint": "arXiv:2310.04153 (v1 6 October 2023; v4 17 April 2025, read in full)", "preprintUrl": "https://arxiv.org/abs/2310.04153", "verified": "arXiv abstract page and v4 full text (HTML); journal, volume, issue, pages and DOI from the Max Planck Institute for Psycholinguistics and Utrecht University publication pages. The journal page itself could not be opened (HTTP 403), so the figures were read in the v4 preprint, whose abstract matches the journal abstract on those pages.", "figures": { "flips": 350757, "people": 48, "coinTypes": 44, "coinTypesNote": "currency × denomination combinations (the paper: '44 different currencies × denominations'; corrected from 46 in the paper's author note). In prose: 'coins of 44 different currencies and denominations', never '44 currencies'.", "sameSide": 0.508, "sameSideInterval95": [ 0.506, 0.509 ], "sameSideNote": "the abstract's figure: the simple binomial result 0.5077 [0.5060, 0.5094] (178,079 / 350,757, 95% credible interval) rounded to three decimals, i.e. the share pooled over all flips, not the average across people", "simpleBinomial": { "sameSide": 0.5077, "sameSideInterval95": [ 0.506, 0.5094 ], "sameSideDisplay": "0.5077 [0.5060, 0.5094]", "heads": 0.5001, "headsInterval95": [ 0.4985, 0.5018 ], "headsDisplay": "0.5001 [0.4985, 0.5018]" }, "hierarchical": { "sameSide": 0.5098, "sameSideInterval95": [ 0.505, 0.5147 ], "sameSideDisplay": "0.5098 [0.5050, 0.5147]", "heads": 0.5005, "headsInterval95": [ 0.4986, 0.5026 ], "headsDisplay": "0.5005 [0.4986, 0.5026]", "note": "The paper's hierarchical model, which lets each person have their own bias: the estimate for a typical flipper (the average across people), as opposed to the pooled share 0.508." }, "sameSideCount": 178079, "headsCount": 175421, "heads": 0.5, "headsNote": "no evidence of a heads or tails bias (abstract: Pr(heads) = 0.500, [0.498, 0.502])", "individualRange": [ 0.487, 0.601 ], "individualRangeNote": "lowest and highest same-side share of a single flipper in the paper's Table 1 (1,600 and 2,800 flips)", "withoutTopFour": 0.506, "withoutTopFourNote": "excluding the four flippers above 53% same-side: 171,517 of 338,985, Pr = 0.5060, [0.5043, 0.5077] (simple binomial)", "withoutTopFourDisplay": "0.5060 [0.5043, 0.5077]", "withoutTopFourHierarchical": 0.506, "withoutTopFourHierarchicalDisplay": "0.5060 [0.5031, 0.5089]", "practiceNote": "around 10,000 flips (about 10 hours of flipping) might be enough to virtually eliminate the bias; the authors add that they cannot rule out other explanations of the decline, such as exhaustion or waning attention. The paper gives no flipping-rate figure." }, "protocol": "Each flip started from the side the coin landed on in the previous flip; coins were caught in the hand, and a flip that was not caught was recorded as a failure and repeated from the same starting position; sequences were videotaped.", "findings": [ "Same-side landings 50.8% overall, with considerable differences between people.", "0.508 is the share pooled over all flips (simple binomial 0.5077 [0.5060, 0.5094]); the hierarchical model's estimate for a typical flipper is 0.5098 [0.5050, 0.5147].", "The bias appears to wane with practice; the authors suggest that around 10,000 flips (about 10 hours) might be enough to virtually eliminate it, while noting that exhaustion or waning attention could also explain the decline.", "Only caught coins were studied: a flip that was not caught was recorded as a failure and repeated, so the paper has no data on coins that land on a table or the floor.", "No heads/tails bias.", "They conclude that when coin flips are used for high-stakes decisions, the starting position of the coin is best concealed." ], "shortQuote": "the starting position of the coin is best concealed", "shortQuoteContext": "from the paper's concluding remarks: when coin flips are used for high-stakes decision-making" }, "dhm": { "title": "Dynamical Bias in the Coin Toss", "authors": "Persi Diaconis, Susan Holmes, Richard Montgomery", "journal": "SIAM Review", "volume": 49, "issue": 2, "pages": "211–235", "year": 2007, "doi": "10.1137/S0036144504446436", "doiUrl": "https://doi.org/10.1137/S0036144504446436", "verified": "full text (PDF, stat.berkeley.edu course copy) read", "findings": [ "Vigorously flipped coins caught in the hand tend to come up the way they started, because the coin precesses (wobbles) as it spins; for natural flips the chance of coming up as started is about .51 (abstract), which section 5.3 says is the mean of their 27 estimates, 0.508, rounded up.", "Estimating p near 1/2 with a standard error of 1/1000 requires about 250,000 flips (a standard error, not a ±0.001 margin; see theory.detection.dhmStandardError).", "A coin caught and then slapped onto the back of the other hand is turned over once, which gives a bias opposite to the start; if the method of catching varies, randomness is added.", "Coins spun on their edge can be far from fair; tossed coins allowed to land and bounce on the floor may spin on edge and may be less fair than coins caught in the hand (the authors' conjecture, not measured as a number). In the same passage they note that people often feel the other way, and suggest that is because coins caught in the hand are easier to manipulate.", "Section 6.5 (No bouncing): their analysis assumes the coin is caught without bouncing; flipped coins that land on hard surfaces and bounce require a different type of analysis.", "Coin tossing by machine can be made to come up as started every time: the toss is physics.", "Section 6.1 (Random In, Random Out): a coin taken from a pocket may be assumed equally likely to start heads up as tails up, and the physics preserves this, so it is then equally likely to end either way; they mention friends who vigorously shake the coin between cupped hands before a toss." ], "shortQuote": "For natural flips, the chance of coming up as started is about .51.", "usefulFlips": 27, "filmedFlips": 50, "meanEstimate": 0.508, "medianEstimate": 0.5027, "sdEstimate": 0.0125, "estimateRange": [ 0.5, 0.545 ], "meanEstimateNote": "Section 5.3 (The Results), just before Table 1: of 50 flips filmed in slow motion, 27 gave useful results; the same-side probabilities computed from them range from 0.500 to 0.545, median 0.5027, standard deviation 0.0125, and the paper says their mean, 0.508, was rounded up to the 0.51 quoted in the abstract. So the 0.51 is the same figure as the 0.508 Bartoš et al. measured, not a separate, larger prediction." }, "adviceSupported": "What the two sources support for a fair hand toss: conceal the starting side (Bartoš et al.); do not let the flipper know the call before the flip (follows from the same-side bias); do not switch between slapping the coin onto the other hand and not slapping it (DHM: slapping reverses the bias). Neither source gives a number or a firm recommendation on catching versus letting the coin land: DHM conjecture that a coin landing on the floor may be less fair, but note that people often feel the other way, that caught coins are easier to manipulate, and that bouncing needs a different type of analysis; Bartoš et al. studied caught coins only." }, "results": { "virtualCoin": { "flips": 20000000, "headsShare": 0.500067, "repeatShare": 0.500124, "pairCounts": { "HH": 5002581, "HT": 4998759, "TH": 4998758, "TT": 4999901 }, "pairSerialStatistic": 1.59, "pairSerialDf": 2, "overlappingPairChiSquare": 1.95, "singleChiSquare": 0.36, "rejectedRedraws": 0, "note": "Simulated spins of a two-entry wheel exactly as app.js spins it (pickWeighted([1, 1]) = below(2)). headsShare = share of the first entry (Yes on the yes or no wheel, Heads on the preset). repeatShare = share of flips equal to the flip before. rejectedRedraws counts rejection-step redraws, which are impossible here (exactly 0). pairSerialStatistic = Good's serial statistic, overlappingPairChiSquare (pair counts over the N - 1 overlapping pairs) minus singleChiSquare (first/second entry counts); for independent fair flips it is chi-square with 2 degrees of freedom (mean 2). The overlapping-pair chi-square on its own is not chi-square with 3 df, because neighboring pairs share a flip." }, "detection": { "note": "Simulated sessions. biased = n hand flips of the 0.508 model (below(1000) < 508), counting same-side landings; virtual = n spins of the two-entry wheel, counting the first entry. flagged = share of sessions the one-sided 5% test flags (same-side count >= flagAt). moreSame/fewerSame = share of sessions with more/fewer same-side than other-side landings.", "rows": [ { "flips": 100, "flagAt": 59, "biased": { "sessions": 200000, "flagged": 0.061065, "moreSame": 0.524885, "fewerSame": 0.397575, "meanSameSide": 50.7946 }, "virtual": { "sessions": 200000, "flagged": 0.04425, "moreFirstEntry": 0.46088 } }, { "flips": 1000, "flagAt": 527, "biased": { "sessions": 40000, "flagged": 0.121175, "moreSame": 0.680625, "fewerSame": 0.297225, "meanSameSide": 508.0151 }, "virtual": { "sessions": 40000, "flagged": 0.047, "moreFirstEntry": 0.4913 } }, { "flips": 24328, "flagAt": 12293, "biased": { "sessions": 3000, "flagged": 0.802 }, "exactPower": 0.801781 } ] }, "studyReplications": { "flips": 350757, "replications": 100, "meanShare": 0.50791, "sdShare": 0.0008817, "lowestShare": 0.506202, "highestShare": 0.510082, "note": "Simulated: the size of the Bartoš et al. experiment (350,757 flips) re-run 100 times with a coin that lands on its starting side exactly 50.8% of the time. lowestShare/highestShare are from this run only (simulation-only; they change on every run and are not checked)." }, "games": { "note": "Simulated with the 0.508 hand-flip model. bestOf rows: the same person sees the starting side and calls it on every toss. hidden: the starting side is drawn at random with below(2) and the caller always calls heads. against: the call is made before the flip and the flipper starts the coin with the called side down.", "bestOf": [ { "bestOf": 1, "matches": 1000000, "pWin": 0.50809, "meanTosses": 1 }, { "bestOf": 3, "matches": 1000000, "pWin": 0.511273, "meanTosses": 2.5001 }, { "bestOf": 5, "matches": 1000000, "pWin": 0.515068, "meanTosses": 4.1264 }, { "bestOf": 7, "matches": 1000000, "pWin": 0.517821, "meanTosses": 5.811 } ], "callHidden": { "tosses": 2000000, "callerWins": 0.500302 }, "callBeforeFlipStartedAgainst": { "tosses": 2000000, "callerWins": 0.492038 } }, "streaks": { "flips": 100, "sessions": 200000, "real": { "meanLongest": 7.1008, "pAtLeast": { "5": 0.977055, "6": 0.82866, "7": 0.572545, "8": 0.33929, "10": 0.097035 } }, "virtual": { "meanLongest": 6.9758, "pAtLeast": { "5": 0.971815, "6": 0.807085, "7": 0.54109, "8": 0.31429, "10": 0.086635 } }, "note": "Simulated sessions of 100 results. real: 99 hand flips of the 0.508 model, each deciding whether the result repeats the one before (the first result's side does not affect run lengths). virtual: 100 spins of the two-entry wheel per session, each compared with the one before." } }, "theory": { "virtualCoin": { "method": "random.js below(2): limit = floor(2^32 / 2) × 2 = 2^32, so no 32-bit draw is ever rejected; index = raw % 2, the last bit of the draw", "limit": 4294967296, "rejectProbability": 0, "pFirstEntry": 0.5, "pFirstEntryFraction": "2147483648/4294967296 = 1/2", "pRepeat": 0.5, "firstEntryRule": "index 0 (Yes / Heads) on an even 32-bit draw, index 1 (No / Tails) on an odd one", "maybePreset": { "entries": 3, "limit": 4294967295, "rejectProbability": 2.3283064365386963e-10, "rejectProbabilityFraction": "1/4294967296", "pEach": "1/3 exactly (after the rejection step)", "note": "The yes or no wheel's 'Add a maybe' preset: below(3) rejects only the single draw 4294967295, then each answer has exactly 1/3." }, "note": "The site's coin has no starting side and no physics: each flip is exactly 1/2 and independent of every other flip. The 2^32 draw count is divisible by 2, so unlike most wheel sizes no draw is ever discarded." }, "detection": { "sameSide": 0.508, "scope": "Every value here is for a coin (or flipper) that lands on its starting side with exactly 0.508 on every flip, independently. 0.508 is the share pooled over all flips in Bartoš et al.; individual flippers ranged from 0.487 to 0.601 (their Table 1), and the bias fell as people flipped more. For other same-side chances, see sampleSizes, widget and chart.", "test": "one-sided exact binomial test at 5%: flag when same-side landings >= flagAt (the smallest count a fair coin reaches with chance <= 5%). Two-sided: 2.5% in each tail; flags a coin that favors either side.", "method": "exact binomial sums; tails at 100 and 1,000 flips verified against exact fractions (crossCheck)", "crossCheck": [ { "flips": 100, "coin": "fair", "atLeast": 50, "exactFraction": 0.539795, "double": 0.539795, "absDiff": 3.3e-16 }, { "flips": 100, "coin": "fair", "atLeast": 51, "exactFraction": 0.460205, "double": 0.460205, "absDiff": 3.9e-16 }, { "flips": 100, "coin": "fair", "atLeast": 59, "exactFraction": 0.044313, "double": 0.044313, "absDiff": 6.9e-18 }, { "flips": 100, "coin": "fair", "atLeast": 61, "exactFraction": 0.0176, "double": 0.0176, "absDiff": 1.7e-17 }, { "flips": 100, "coin": "0.508", "atLeast": 50, "exactFraction": 0.602695, "double": 0.602695, "absDiff": 2.2e-16 }, { "flips": 100, "coin": "0.508", "atLeast": 51, "exactFraction": 0.524118, "double": 0.524118, "absDiff": 2.2e-16 }, { "flips": 100, "coin": "0.508", "atLeast": 59, "exactFraction": 0.061451, "double": 0.061451, "absDiff": 7.6e-17 }, { "flips": 100, "coin": "0.508", "atLeast": 61, "exactFraction": 0.025811, "double": 0.025811, "absDiff": 1e-16 }, { "flips": 1000, "coin": "fair", "atLeast": 500, "exactFraction": 0.512613, "double": 0.512613, "absDiff": 0 }, { "flips": 1000, "coin": "fair", "atLeast": 501, "exactFraction": 0.487387, "double": 0.487387, "absDiff": 5.6e-17 }, { "flips": 1000, "coin": "fair", "atLeast": 527, "exactFraction": 0.046844, "double": 0.046844, "absDiff": 2.8e-17 }, { "flips": 1000, "coin": "fair", "atLeast": 532, "exactFraction": 0.023146, "double": 0.023146, "absDiff": 3.5e-18 }, { "flips": 1000, "coin": "0.508", "atLeast": 500, "exactFraction": 0.704623, "double": 0.704623, "absDiff": 1.1e-15 }, { "flips": 1000, "coin": "0.508", "atLeast": 501, "exactFraction": 0.682429, "double": 0.682429, "absDiff": 1e-15 }, { "flips": 1000, "coin": "0.508", "atLeast": 527, "exactFraction": 0.120942, "double": 0.120942, "absDiff": 6.9e-17 }, { "flips": 1000, "coin": "0.508", "atLeast": 532, "exactFraction": 0.068532, "double": 0.068532, "absDiff": 8.3e-17 } ], "sampleSizes": [ { "sameSide": 0.505, "edgePoints": "0.50 points", "oneSided80": { "approx": 61824, "firstReached": 61826, "staysAbove": 62186, "checkedTo": 72598, "powerAtStaysAbove": 0.801116 }, "oneSided90": { "approx": 85635, "firstReached": 85721, "staysAbove": 85976, "checkedTo": 99981, "powerAtStaysAbove": 0.9006 }, "twoSided80": { "approx": 78487, "firstReached": 78595, "staysAbove": 78828, "checkedTo": 91761, "powerAtStaysAbove": 0.800993 }, "twoSided90": { "approx": 105071, "firstReached": 105157, "staysAbove": 105406, "checkedTo": 122332, "powerAtStaysAbove": 0.900543 } }, { "sameSide": 0.508, "edgePoints": "0.80 points", "oneSided80": { "approx": 24149, "firstReached": 24179, "staysAbove": 24328, "checkedTo": 29272, "powerAtStaysAbove": 0.801781 }, "oneSided90": { "approx": 33449, "firstReached": 33464, "staysAbove": 33623, "checkedTo": 39967, "powerAtStaysAbove": 0.900956 }, "twoSided80": { "approx": 30658, "firstReached": 30719, "staysAbove": 30864, "checkedTo": 36757, "powerAtStaysAbove": 0.80157 }, "twoSided90": { "approx": 41041, "firstReached": 41115, "staysAbove": 41270, "checkedTo": 48698, "powerAtStaysAbove": 0.900864 } }, { "sameSide": 0.51, "edgePoints": "1.00 points", "oneSided80": { "approx": 15455, "firstReached": 15486, "staysAbove": 15605, "checkedTo": 19274, "powerAtStaysAbove": 0.802206 }, "oneSided90": { "approx": 21406, "firstReached": 21430, "staysAbove": 21557, "checkedTo": 26117, "powerAtStaysAbove": 0.901205 }, "twoSided80": { "approx": 19620, "firstReached": 19648, "staysAbove": 19765, "checkedTo": 24063, "powerAtStaysAbove": 0.801994 }, "twoSided90": { "approx": 26265, "firstReached": 26289, "staysAbove": 26412, "checkedTo": 31705, "powerAtStaysAbove": 0.901079 } }, { "sameSide": 0.515, "edgePoints": "1.50 points", "oneSided80": { "approx": 6868, "firstReached": 6896, "staysAbove": 6975, "checkedTo": 9399, "powerAtStaysAbove": 0.80332 }, "oneSided90": { "approx": 9512, "firstReached": 9538, "staysAbove": 9621, "checkedTo": 12439, "powerAtStaysAbove": 0.901739 }, "twoSided80": { "approx": 8719, "firstReached": 8759, "staysAbove": 8836, "checkedTo": 11527, "powerAtStaysAbove": 0.802984 }, "twoSided90": { "approx": 11671, "firstReached": 11683, "staysAbove": 11764, "checkedTo": 14922, "powerAtStaysAbove": 0.901572 } }, { "sameSide": 0.52, "edgePoints": "2.00 points", "oneSided80": { "approx": 3863, "firstReached": 3886, "staysAbove": 3945, "checkedTo": 5943, "powerAtStaysAbove": 0.80436 }, "oneSided90": { "approx": 5349, "firstReached": 5372, "staysAbove": 5435, "checkedTo": 7652, "powerAtStaysAbove": 0.902335 }, "twoSided80": { "approx": 4904, "firstReached": 4927, "staysAbove": 4984, "checkedTo": 7140, "powerAtStaysAbove": 0.803924 }, "twoSided90": { "approx": 6563, "firstReached": 6572, "staysAbove": 6633, "checkedTo": 9048, "powerAtStaysAbove": 0.902186 } }, { "sameSide": 0.53, "edgePoints": "3.00 points", "oneSided80": { "approx": 1716, "firstReached": 1734, "staysAbove": 1773, "checkedTo": 3474, "powerAtStaysAbove": 0.806661 }, "oneSided90": { "approx": 2376, "firstReached": 2394, "staysAbove": 2435, "checkedTo": 4233, "powerAtStaysAbove": 0.903387 }, "twoSided80": { "approx": 2178, "firstReached": 2189, "staysAbove": 2228, "checkedTo": 4005, "powerAtStaysAbove": 0.806066 }, "twoSided90": { "approx": 2915, "firstReached": 2919, "staysAbove": 2960, "checkedTo": 4853, "powerAtStaysAbove": 0.90316 } }, { "sameSide": 0.55, "edgePoints": "5.00 points", "oneSided80": { "approx": 617, "firstReached": 620, "staysAbove": 654, "checkedTo": 2210, "powerAtStaysAbove": 0.810731 }, "oneSided90": { "approx": 853, "firstReached": 866, "staysAbove": 891, "checkedTo": 2481, "powerAtStaysAbove": 0.905874 }, "twoSided80": { "approx": 783, "firstReached": 786, "staysAbove": 820, "checkedTo": 2401, "powerAtStaysAbove": 0.809944 }, "twoSided90": { "approx": 1047, "firstReached": 1055, "staysAbove": 1080, "checkedTo": 2705, "powerAtStaysAbove": 0.90564 } }, { "sameSide": 0.6, "edgePoints": "10.00 points", "oneSided80": { "approx": 153, "firstReached": 158, "staysAbove": 169, "checkedTo": 1676, "powerAtStaysAbove": 0.823068 }, "oneSided90": { "approx": 211, "firstReached": 213, "staysAbove": 224, "checkedTo": 1743, "powerAtStaysAbove": 0.911006 }, "twoSided80": { "approx": 194, "firstReached": 199, "staysAbove": 210, "checkedTo": 1724, "powerAtStaysAbove": 0.820248 }, "twoSided90": { "approx": 259, "firstReached": 263, "staysAbove": 274, "checkedTo": 1798, "powerAtStaysAbove": 0.910076 } } ], "sampleSize508": { "oneSided80": 24328, "oneSided90": 33623, "twoSided80": 30864, "twoSided90": 41270, "oneSided80Display": "24,328", "twoSided90Display": "41,270", "oneSided80Rounded": "about 24,000", "rangeRounded": "roughly 24,000 to 41,000", "oneSided80Variants": { "normalApprox": 24149, "firstReached": 24179, "staysAbove": 24328 }, "note": "Quote these in prose as rounded figures (oneSided80Rounded; rangeRounded runs from one-sided 80% to two-sided 90%), not as precise thresholds: exact power is a sawtooth in the number of flips (the flag count moves in whole steps), so 'how many flips' has three defensible answers (oneSided80Variants: textbook normal approximation, first n that reaches 80%, n from which it stays at or above 80%). For the same reason a few more flips do not always raise power (see the table: 200 flips is slightly below 100); avoid comparing nearby flip counts. All values assume a coin (or flipper) with exactly a 50.8% same-side chance on every flip.", "hoursAtPapersPace": { "oneSided80": 24.328, "twoSided90": 41.27, "note": "Our own arithmetic, not a figure the paper gives: flips / 1,000, the rate implied by Bartoš et al.'s aside that around 10,000 flips take about 10 hours. The paper makes that remark about the bias fading with practice (around 10,000 flips might be enough to virtually eliminate it) and adds that it cannot rule out other explanations of the decline, such as exhaustion or waning attention. So one person flipping this long may not keep a 50.8% bias; present the hours as a rough scale only." } }, "table": [ { "flips": 10, "expectedSameSide": 5.08, "extraSameSideOverFair": 0.08, "sdOfCount": 1.5809, "edgeInSds": 0.0506, "oneSided": { "flagAt": 9, "flagAtShare": 0.9, "falseFlag": 0.010742, "power": 0.01223, "powerDisplay": "1.22%", "missChance": 0.9878, "powerOverFalseFlag": 1.1385 }, "twoSided": { "flagAt": 9, "flagLowAt": 1, "falseFlag": 0.021484, "power": 0.021642, "powerDisplay": "2.16%" }, "pMoreSameThanOther": 0.396791, "pMoreSameThanOtherDisplay": "39.7%", "pMoreSameThanOtherFair": 0.376953, "pMoreSameThanOtherFairDisplay": "37.7%", "pTie": 0.245779, "pFewerSameThanOther": 0.35743, "pFewerSameThanOtherDisplay": "35.7%", "seOfShare": 0.1581 }, { "flips": 20, "expectedSameSide": 10.16, "extraSameSideOverFair": 0.16, "sdOfCount": 2.2358, "edgeInSds": 0.0716, "oneSided": { "flagAt": 15, "flagAtShare": 0.75, "falseFlag": 0.020695, "power": 0.024508, "powerDisplay": "2.45%", "missChance": 0.9755, "powerOverFalseFlag": 1.1843 }, "twoSided": { "flagAt": 15, "flagLowAt": 5, "falseFlag": 0.041389, "power": 0.041901, "powerDisplay": "4.19%" }, "pMoreSameThanOther": 0.440297, "pMoreSameThanOtherDisplay": "44.0%", "pMoreSameThanOtherFair": 0.411901, "pMoreSameThanOtherFairDisplay": "41.2%", "pTie": 0.175747, "pFewerSameThanOther": 0.383957, "pFewerSameThanOtherDisplay": "38.4%", "seOfShare": 0.1118 }, { "flips": 50, "expectedSameSide": 25.4, "extraSameSideOverFair": 0.4, "sdOfCount": 3.5351, "edgeInSds": 0.1132, "oneSided": { "flagAt": 32, "flagAtShare": 0.64, "falseFlag": 0.032454, "power": 0.04156, "powerDisplay": "4.16%", "missChance": 0.9584, "powerOverFalseFlag": 1.2806 }, "twoSided": { "flagAt": 33, "flagLowAt": 17, "falseFlag": 0.032839, "power": 0.033949, "powerDisplay": "3.39%" }, "pMoreSameThanOther": 0.489039, "pMoreSameThanOtherDisplay": "48.9%", "pMoreSameThanOtherFair": 0.443862, "pMoreSameThanOtherFairDisplay": "44.4%", "pTie": 0.111559, "pFewerSameThanOther": 0.399402, "pFewerSameThanOtherDisplay": "39.9%", "seOfShare": 0.0707 }, { "flips": 100, "expectedSameSide": 50.8, "extraSameSideOverFair": 0.8, "sdOfCount": 4.9994, "edgeInSds": 0.16, "oneSided": { "flagAt": 59, "flagAtShare": 0.59, "falseFlag": 0.044313, "power": 0.061451, "powerDisplay": "6.15%", "missChance": 0.9385, "powerOverFalseFlag": 1.3868 }, "twoSided": { "flagAt": 61, "flagLowAt": 39, "falseFlag": 0.0352, "power": 0.037539, "powerDisplay": "3.75%" }, "pMoreSameThanOther": 0.524118, "pMoreSameThanOtherDisplay": "52.4%", "pMoreSameThanOtherFair": 0.460205, "pMoreSameThanOtherFairDisplay": "46.0%", "pTie": 0.078577, "pFewerSameThanOther": 0.397305, "pFewerSameThanOtherDisplay": "39.7%", "seOfShare": 0.04999 }, { "flips": 200, "expectedSameSide": 101.6, "extraSameSideOverFair": 1.6, "sdOfCount": 7.0702, "edgeInSds": 0.2263, "oneSided": { "flagAt": 113, "flagAtShare": 0.565, "falseFlag": 0.038419, "power": 0.061403, "powerDisplay": "6.14%", "missChance": 0.9386, "powerOverFalseFlag": 1.5983 }, "twoSided": { "flagAt": 115, "flagLowAt": 85, "falseFlag": 0.040037, "power": 0.045148, "powerDisplay": "4.51%" }, "pMoreSameThanOther": 0.56194, "pMoreSameThanOtherDisplay": "56.2%", "pMoreSameThanOtherFair": 0.471826, "pMoreSameThanOtherFairDisplay": "47.2%", "pTie": 0.054924, "pFewerSameThanOther": 0.383136, "pFewerSameThanOtherDisplay": "38.3%", "seOfShare": 0.03535 }, { "flips": 500, "expectedSameSide": 254, "extraSameSideOverFair": 4, "sdOfCount": 11.1789, "edgeInSds": 0.3578, "oneSided": { "flagAt": 269, "flagAtShare": 0.538, "falseFlag": 0.048945, "power": 0.097247, "powerDisplay": "9.72%", "missChance": 0.9028, "powerOverFalseFlag": 1.9869 }, "twoSided": { "flagAt": 273, "flagLowAt": 227, "falseFlag": 0.044063, "power": 0.057745, "powerDisplay": "5.77%" }, "pMoreSameThanOther": 0.622958, "pMoreSameThanOtherDisplay": "62.3%", "pMoreSameThanOtherFair": 0.482168, "pMoreSameThanOtherFairDisplay": "48.2%", "pTie": 0.033453, "pFewerSameThanOther": 0.343589, "pFewerSameThanOtherDisplay": "34.4%", "seOfShare": 0.02236 }, { "flips": 1000, "expectedSameSide": 508, "extraSameSideOverFair": 8, "sdOfCount": 15.8094, "edgeInSds": 0.506, "oneSided": { "flagAt": 527, "flagAtShare": 0.527, "falseFlag": 0.046844, "power": 0.120942, "powerDisplay": "12.1%", "missChance": 0.8791, "powerOverFalseFlag": 2.5818 }, "twoSided": { "flagAt": 532, "flagLowAt": 468, "falseFlag": 0.046291, "power": 0.074764, "powerDisplay": "7.48%" }, "pMoreSameThanOther": 0.682429, "pMoreSameThanOtherDisplay": "68.2%", "pMoreSameThanOtherFair": 0.487387, "pMoreSameThanOtherFairDisplay": "48.7%", "pTie": 0.022194, "pFewerSameThanOther": 0.295377, "pFewerSameThanOtherDisplay": "29.5%", "seOfShare": 0.01581 }, { "flips": 2000, "expectedSameSide": 1016, "extraSameSideOverFair": 16, "sdOfCount": 22.3578, "edgeInSds": 0.7156, "oneSided": { "flagAt": 1038, "flagAtShare": 0.519, "falseFlag": 0.046754, "power": 0.16812, "powerDisplay": "16.8%", "missChance": 0.8319, "powerOverFalseFlag": 3.5959 }, "twoSided": { "flagAt": 1045, "flagLowAt": 955, "falseFlag": 0.046553, "power": 0.10459, "powerDisplay": "10.5%" }, "pMoreSameThanOther": 0.755945, "pMoreSameThanOtherDisplay": "75.6%", "pMoreSameThanOtherFair": 0.49108, "pMoreSameThanOtherFairDisplay": "49.1%", "pTie": 0.013809, "pFewerSameThanOther": 0.230246, "pFewerSameThanOtherDisplay": "23.0%", "seOfShare": 0.01118 }, { "flips": 5000, "expectedSameSide": 2540, "extraSameSideOverFair": 40, "sdOfCount": 35.3508, "edgeInSds": 1.1315, "oneSided": { "flagAt": 2559, "flagAtShare": 0.5118, "falseFlag": 0.048995, "power": 0.300395, "powerDisplay": "30.0%", "missChance": 0.6996, "powerOverFalseFlag": 6.1311 }, "twoSided": { "flagAt": 2570, "flagLowAt": 2430, "falseFlag": 0.049316, "power": 0.202986, "powerDisplay": "20.3%" }, "pMoreSameThanOther": 0.868079, "pMoreSameThanOtherDisplay": "86.8%", "pMoreSameThanOtherFair": 0.494358, "pMoreSameThanOtherFairDisplay": "49.4%", "pTie": 0.005949, "pFewerSameThanOther": 0.125972, "pFewerSameThanOtherDisplay": "12.6%", "seOfShare": 0.00707 }, { "flips": 10000, "expectedSameSide": 5080, "extraSameSideOverFair": 80, "sdOfCount": 49.9936, "edgeInSds": 1.6002, "oneSided": { "flagAt": 5083, "flagAtShare": 0.5083, "falseFlag": 0.049469, "power": 0.48008, "powerDisplay": "48.0%", "missChance": 0.5199, "powerOverFalseFlag": 9.7047 }, "twoSided": { "flagAt": 5099, "flagLowAt": 4901, "falseFlag": 0.048833, "power": 0.35587, "powerDisplay": "35.6%" }, "pMoreSameThanOther": 0.944099, "pMoreSameThanOtherDisplay": "94.41%", "pMoreSameThanOtherFair": 0.496011, "pMoreSameThanOtherFairDisplay": "49.6%", "pTie": 0.002218, "pFewerSameThanOther": 0.053683, "pFewerSameThanOtherDisplay": "5.37%", "seOfShare": 0.004999 }, { "flips": 20000, "expectedSameSide": 10160, "extraSameSideOverFair": 160, "sdOfCount": 70.7016, "edgeInSds": 2.263, "oneSided": { "flagAt": 10117, "flagAtShare": 0.50585, "falseFlag": 0.04972, "power": 0.730816, "powerDisplay": "73.1%", "missChance": 0.2692, "powerOverFalseFlag": 14.6985 }, "twoSided": { "flagAt": 10140, "flagLowAt": 9860, "falseFlag": 0.048512, "power": 0.614097, "powerDisplay": "61.4%" }, "pMoreSameThanOther": 0.98796, "pMoreSameThanOtherDisplay": "98.80%", "pMoreSameThanOtherFair": 0.497179, "pMoreSameThanOtherFairDisplay": "49.7%", "pTie": 0.000436, "pFewerSameThanOther": 0.011604, "pFewerSameThanOtherDisplay": "1.16%", "seOfShare": 0.003535 }, { "flips": 50000, "expectedSameSide": 25400, "extraSameSideOverFair": 400, "sdOfCount": 111.7891, "edgeInSds": 3.5782, "oneSided": { "flagAt": 25185, "flagAtShare": 0.5037, "falseFlag": 0.049449, "power": 0.973053, "powerDisplay": "97.31%", "missChance": 0.02695, "powerOverFalseFlag": 19.678 }, "twoSided": { "flagAt": 25220, "flagLowAt": 24780, "falseFlag": 0.049614, "power": 0.946803, "powerDisplay": "94.68%" }, "pMoreSameThanOther": 0.999824, "pMoreSameThanOtherDisplay": "99.98%", "pMoreSameThanOtherFair": 0.498216, "pMoreSameThanOtherFairDisplay": "49.8%", "pTie": 0.000006, "pFewerSameThanOther": 0.00017, "pFewerSameThanOtherDisplay": "0.02%", "seOfShare": 0.002236 }, { "flips": 100000, "expectedSameSide": 50800, "extraSameSideOverFair": 800, "sdOfCount": 158.0936, "edgeInSds": 5.0603, "oneSided": { "flagAt": 50261, "flagAtShare": 0.50261, "falseFlag": 0.049723, "power": 0.999678, "powerDisplay": "99.97%", "missChance": 0.000322, "powerOverFalseFlag": 20.1051 }, "twoSided": { "flagAt": 50311, "flagLowAt": 49689, "falseFlag": 0.049556, "power": 0.99902, "powerDisplay": "99.90%" }, "pMoreSameThanOther": 1, "pMoreSameThanOtherDisplay": ">99.99%", "pMoreSameThanOtherFair": 0.498738, "pMoreSameThanOtherFairDisplay": "49.9%", "pTie": 0, "pFewerSameThanOther": 0, "pFewerSameThanOtherDisplay": "<0.01%", "seOfShare": 0.001581 }, { "flips": 200000, "expectedSameSide": 101600, "extraSameSideOverFair": 1600, "sdOfCount": 223.5782, "edgeInSds": 7.1563, "oneSided": { "flagAt": 100369, "flagAtShare": 0.501845, "falseFlag": 0.049678, "power": 1, "powerDisplay": ">99.99%", "missChance": 1.816e-8, "powerOverFalseFlag": 20.1296 }, "twoSided": { "flagAt": 100439, "flagLowAt": 99561, "falseFlag": 0.049875, "power": 1, "powerDisplay": ">99.99%" }, "pMoreSameThanOther": 1, "pMoreSameThanOtherDisplay": ">99.99%", "pMoreSameThanOtherFair": 0.499108, "pMoreSameThanOtherFairDisplay": "49.9%", "pTie": 0, "pFewerSameThanOther": 0, "pFewerSameThanOtherDisplay": "<0.01%", "seOfShare": 0.001118 }, { "flips": 350757, "expectedSameSide": 178184.556, "extraSameSideOverFair": 2806.056, "sdOfCount": 296.0858, "edgeInSds": 9.4772, "oneSided": { "flagAt": 175867, "flagAtShare": 0.501393, "falseFlag": 0.04968, "power": 1, "powerDisplay": ">99.99%", "missChance": 2.467e-15, "powerOverFalseFlag": 20.1287 }, "twoSided": { "flagAt": 175960, "flagLowAt": 174797, "falseFlag": 0.04976, "power": 1, "powerDisplay": ">99.99%" }, "pMoreSameThanOther": 1, "pMoreSameThanOtherDisplay": ">99.99%", "pMoreSameThanOtherFair": 0.5, "pMoreSameThanOtherFairDisplay": "50.0%", "pTie": 0, "pFewerSameThanOther": 0, "pFewerSameThanOtherDisplay": "<0.01%", "seOfShare": 0.0008441 }, { "flips": 500000, "expectedSameSide": 254000, "extraSameSideOverFair": 4000, "sdOfCount": 353.5081, "edgeInSds": 11.3152, "oneSided": { "flagAt": 250583, "flagAtShare": 0.501166, "falseFlag": 0.049722, "power": 1, "powerDisplay": ">99.99%", "missChance": 2.085e-22, "powerOverFalseFlag": 20.112 }, "twoSided": { "flagAt": 250694, "flagLowAt": 249306, "falseFlag": 0.049819, "power": 1, "powerDisplay": ">99.99%" }, "pMoreSameThanOther": 1, "pMoreSameThanOtherDisplay": ">99.99%", "pMoreSameThanOtherFair": 0.499436, "pMoreSameThanOtherFairDisplay": "49.9%", "pTie": 0, "pFewerSameThanOther": 0, "pFewerSameThanOtherDisplay": "<0.01%", "seOfShare": 0.000707 }, { "flips": 1000000, "expectedSameSide": 508000, "extraSameSideOverFair": 8000, "sdOfCount": 499.936, "edgeInSds": 16.002, "oneSided": { "flagAt": 500823, "flagAtShare": 0.500823, "falseFlag": 0.049985, "power": 1, "powerDisplay": ">99.99%", "missChance": 4.886e-47, "powerOverFalseFlag": 20.006 }, "twoSided": { "flagAt": 500981, "flagLowAt": 499019, "falseFlag": 0.049879, "power": 1, "powerDisplay": ">99.99%" }, "pMoreSameThanOther": 1, "pMoreSameThanOtherDisplay": ">99.99%", "pMoreSameThanOtherFair": 0.499601, "pMoreSameThanOtherFairDisplay": "50.0%", "pTie": 0, "pFewerSameThanOther": 0, "pFewerSameThanOtherDisplay": "<0.01%", "seOfShare": 0.0004999 } ], "at100": { "flips": 100, "expectedSameSide": 50.8, "extraSameSideOverFair": 0.8, "sdOfCount": 4.9994, "edgeInSds": 0.16, "oneSided": { "flagAt": 59, "flagAtShare": 0.59, "falseFlag": 0.044313, "power": 0.061451, "powerDisplay": "6.15%", "missChance": 0.9385, "powerOverFalseFlag": 1.3868 }, "twoSided": { "flagAt": 61, "flagLowAt": 39, "falseFlag": 0.0352, "power": 0.037539, "powerDisplay": "3.75%" }, "pMoreSameThanOther": 0.524118, "pMoreSameThanOtherDisplay": "52.4%", "pMoreSameThanOtherFair": 0.460205, "pMoreSameThanOtherFairDisplay": "46.0%", "pTie": 0.078577, "pFewerSameThanOther": 0.397305, "pFewerSameThanOtherDisplay": "39.7%", "seOfShare": 0.04999 }, "at1000": { "flips": 1000, "expectedSameSide": 508, "extraSameSideOverFair": 8, "sdOfCount": 15.8094, "edgeInSds": 0.506, "oneSided": { "flagAt": 527, "flagAtShare": 0.527, "falseFlag": 0.046844, "power": 0.120942, "powerDisplay": "12.1%", "missChance": 0.8791, "powerOverFalseFlag": 2.5818 }, "twoSided": { "flagAt": 532, "flagLowAt": 468, "falseFlag": 0.046291, "power": 0.074764, "powerDisplay": "7.48%" }, "pMoreSameThanOther": 0.682429, "pMoreSameThanOtherDisplay": "68.2%", "pMoreSameThanOtherFair": 0.487387, "pMoreSameThanOtherFairDisplay": "48.7%", "pTie": 0.022194, "pFewerSameThanOther": 0.295377, "pFewerSameThanOtherDisplay": "29.5%", "seOfShare": 0.01581 }, "dhmStandardError": { "note": "Diaconis, Holmes & Montgomery (2007): estimating p near 1/2 with a STANDARD ERROR of 1/1000 needs n = 250,000 flips, since SE = 1/(2 sqrt(n)). Recomputed here. A standard error of 0.001 is not a margin of ±0.001: the 95% margin at 250,000 flips is about ±0.002 (margin95At250000), and a 95% margin of ±0.001 needs about 960,000 flips (flipsForMargin95Of001). Their conclusion also says that detecting departures of the size they found would require 250,000 tosses.", "flipsForSe001": 250000, "sePerFlipsAt350757": 0.0008442, "margin95At250000": 0.00196, "flipsForMargin95Of001": 960365 }, "chart": [ { "flips": 10, "flagAt": 9, "falseFlag": 0.010742, "power508": 0.01223, "power51": 0.012628, "power52": 0.014789, "power55": 0.023257 }, { "flips": 11, "flagAt": 9, "falseFlag": 0.032715, "power508": 0.036772, "power51": 0.037847, "power52": 0.043611, "power55": 0.065224 }, { "flips": 13, "flagAt": 10, "falseFlag": 0.046143, "power508": 0.052002, "power51": 0.053555, "power52": 0.061873, "power55": 0.092921 }, { "flips": 14, "flagAt": 11, "falseFlag": 0.028687, "power508": 0.032822, "power51": 0.033928, "power52": 0.039926, "power55": 0.063215 }, { "flips": 16, "flagAt": 12, "falseFlag": 0.038406, "power508": 0.044045, "power51": 0.045553, "power52": 0.053725, "power55": 0.085309 }, { "flips": 18, "flagAt": 13, "falseFlag": 0.048126, "power508": 0.055315, "power51": 0.057237, "power52": 0.067646, "power55": 0.107697 }, { "flips": 20, "flagAt": 15, "falseFlag": 0.020695, "power508": 0.024508, "power51": 0.025548, "power52": 0.031315, "power55": 0.055334 }, { "flips": 22, "flagAt": 16, "falseFlag": 0.026239, "power508": 0.031133, "power51": 0.032467, "power52": 0.039861, "power55": 0.070515 }, { "flips": 25, "flagAt": 18, "falseFlag": 0.021643, "power508": 0.026112, "power51": 0.027342, "power52": 0.034237, "power55": 0.063851 }, { "flips": 28, "flagAt": 19, "falseFlag": 0.043579, "power508": 0.051982, "power51": 0.054272, "power52": 0.066935, "power55": 0.118721 }, { "flips": 32, "flagAt": 22, "falseFlag": 0.025051, "power508": 0.030824, "power51": 0.032428, "power52": 0.041505, "power55": 0.081527 }, { "flips": 35, "flagAt": 23, "falseFlag": 0.044766, "power508": 0.054443, "power51": 0.057103, "power52": 0.071967, "power55": 0.134357 }, { "flips": 40, "flagAt": 26, "falseFlag": 0.040345, "power508": 0.049929, "power51": 0.052588, "power52": 0.067605, "power55": 0.132602 }, { "flips": 45, "flagAt": 29, "falseFlag": 0.036227, "power508": 0.04561, "power51": 0.048237, "power52": 0.063235, "power55": 0.130182 }, { "flips": 50, "flagAt": 32, "falseFlag": 0.032454, "power508": 0.04156, "power51": 0.044134, "power52": 0.058983, "power55": 0.127345 }, { "flips": 56, "flagAt": 35, "falseFlag": 0.040713, "power508": 0.052319, "power51": 0.055597, "power52": 0.074473, "power55": 0.160172 }, { "flips": 63, "flagAt": 39, "falseFlag": 0.038463, "power508": 0.050309, "power51": 0.053684, "power52": 0.073323, "power55": 0.164854 }, { "flips": 71, "flagAt": 43, "falseFlag": 0.047962, "power508": 0.062994, "power51": 0.067273, "power52": 0.092106, "power55": 0.205716 }, { "flips": 79, "flagAt": 48, "falseFlag": 0.035582, "power508": 0.048227, "power51": 0.051894, "power52": 0.073655, "power55": 0.180061 }, { "flips": 89, "flagAt": 53, "falseFlag": 0.04466, "power508": 0.060804, "power51": 0.06548, "power52": 0.093139, "power55": 0.225262 }, { "flips": 100, "flagAt": 59, "falseFlag": 0.044313, "power508": 0.061451, "power51": 0.066457, "power52": 0.096335, "power55": 0.241491 }, { "flips": 112, "flagAt": 66, "falseFlag": 0.036063, "power508": 0.051651, "power51": 0.056286, "power52": 0.084543, "power55": 0.229939 }, { "flips": 126, "flagAt": 73, "falseFlag": 0.045061, "power508": 0.064866, "power51": 0.070745, "power52": 0.106426, "power55": 0.284083 }, { "flips": 141, "flagAt": 81, "falseFlag": 0.045885, "power508": 0.067307, "power51": 0.073716, "power52": 0.112918, "power55": 0.309588 }, { "flips": 158, "flagAt": 90, "falseFlag": 0.047237, "power508": 0.070627, "power51": 0.077679, "power52": 0.121116, "power55": 0.339666 }, { "flips": 178, "flagAt": 101, "falseFlag": 0.042217, "power508": 0.065258, "power51": 0.072322, "power52": 0.116657, "power55": 0.348479 }, { "flips": 200, "flagAt": 113, "falseFlag": 0.038419, "power508": 0.061403, "power51": 0.068568, "power52": 0.114354, "power55": 0.361997 }, { "flips": 224, "flagAt": 125, "falseFlag": 0.047312, "power508": 0.076067, "power51": 0.085006, "power52": 0.141687, "power55": 0.431579 }, { "flips": 251, "flagAt": 140, "falseFlag": 0.038481, "power508": 0.064869, "power51": 0.073282, "power52": 0.128212, "power55": 0.427846 }, { "flips": 282, "flagAt": 156, "falseFlag": 0.042, "power508": 0.072245, "power51": 0.081945, "power52": 0.145442, "power55": 0.481706 }, { "flips": 316, "flagAt": 174, "falseFlag": 0.040508, "power508": 0.072086, "power51": 0.082367, "power52": 0.150632, "power55": 0.514279 }, { "flips": 355, "flagAt": 194, "falseFlag": 0.044647, "power508": 0.081104, "power51": 0.093031, "power52": 0.172232, "power55": 0.574707 }, { "flips": 398, "flagAt": 216, "falseFlag": 0.048988, "power508": 0.09085, "power51": 0.104606, "power52": 0.195832, "power55": 0.634573 }, { "flips": 447, "flagAt": 242, "falseFlag": 0.044252, "power508": 0.086115, "power51": 0.100184, "power52": 0.195578, "power55": 0.660863 }, { "flips": 501, "flagAt": 270, "falseFlag": 0.044731, "power508": 0.090102, "power51": 0.105542, "power52": 0.211052, "power55": 0.706884 }, { "flips": 562, "flagAt": 301, "falseFlag": 0.04993, "power508": 0.102717, "power51": 0.120721, "power52": 0.24288, "power55": 0.767246 }, { "flips": 631, "flagAt": 337, "falseFlag": 0.047224, "power508": 0.101958, "power51": 0.121003, "power52": 0.25226, "power55": 0.800822 }, { "flips": 708, "flagAt": 377, "falseFlag": 0.045364, "power508": 0.102785, "power51": 0.123146, "power52": 0.265321, "power55": 0.835113 }, { "flips": 794, "flagAt": 421, "falseFlag": 0.04763, "power508": 0.111721, "power51": 0.134643, "power52": 0.294286, "power55": 0.87601 }, { "flips": 891, "flagAt": 471, "falseFlag": 0.046933, "power508": 0.115507, "power51": 0.140441, "power52": 0.315228, "power55": 0.905874 }, { "flips": 1000, "flagAt": 527, "falseFlag": 0.046844, "power508": 0.120942, "power51": 0.148292, "power52": 0.340519, "power55": 0.932231 }, { "flips": 1122, "flagAt": 590, "falseFlag": 0.044385, "power508": 0.121811, "power51": 0.151041, "power52": 0.358769, "power55": 0.95101 }, { "flips": 1259, "flagAt": 660, "falseFlag": 0.0454, "power508": 0.13062, "power51": 0.163168, "power52": 0.392989, "power55": 0.96888 }, { "flips": 1413, "flagAt": 738, "falseFlag": 0.049518, "power508": 0.147297, "power51": 0.184668, "power52": 0.44214, "power55": 0.982894 }, { "flips": 1585, "flagAt": 826, "falseFlag": 0.048665, "power508": 0.153643, "power51": 0.194436, "power52": 0.474078, "power55": 0.990148 }, { "flips": 1778, "flagAt": 925, "falseFlag": 0.046095, "power508": 0.156421, "power51": 0.200289, "power52": 0.501262, "power55": 0.994487 }, { "flips": 1995, "flagAt": 1035, "falseFlag": 0.048772, "power508": 0.17304, "power51": 0.222573, "power52": 0.551814, "power55": 0.997595 }, { "flips": 2239, "flagAt": 1159, "falseFlag": 0.049623, "power508": 0.186354, "power51": 0.241304, "power52": 0.596677, "power55": 0.999012 }, { "flips": 2512, "flagAt": 1298, "falseFlag": 0.048848, "power508": 0.196501, "power51": 0.256661, "power52": 0.636556, "power55": 0.99962 }, { "flips": 2818, "flagAt": 1454, "falseFlag": 0.046805, "power508": 0.204042, "power51": 0.269311, "power52": 0.6727, "power55": 0.999866 }, { "flips": 3162, "flagAt": 1628, "falseFlag": 0.049069, "power508": 0.225361, "power51": 0.298316, "power52": 0.724417, "power55": 0.999966 }, { "flips": 3548, "flagAt": 1824, "falseFlag": 0.048245, "power508": 0.239149, "power51": 0.318912, "power52": 0.764617, "power55": 0.999992 }, { "flips": 3981, "flagAt": 2043, "falseFlag": 0.049639, "power508": 0.261475, "power51": 0.349607, "power52": 0.809553, "power55": 0.999999 }, { "flips": 4467, "flagAt": 2289, "falseFlag": 0.049894, "power508": 0.282148, "power51": 0.378629, "power52": 0.848121, "power55": 1 }, { "flips": 5012, "flagAt": 2565, "falseFlag": 0.049196, "power508": 0.301558, "power51": 0.406447, "power52": 0.88101, "power55": 1 }, { "flips": 5623, "flagAt": 2874, "falseFlag": 0.049097, "power508": 0.324973, "power51": 0.43887, "power52": 0.910976, "power55": 1 }, { "flips": 6310, "flagAt": 3221, "falseFlag": 0.049556, "power508": 0.352658, "power51": 0.475937, "power52": 0.936906, "power55": 1 }, { "flips": 7079, "flagAt": 3610, "falseFlag": 0.048057, "power508": 0.375338, "power51": 0.507524, "power52": 0.95568, "power55": 1 }, { "flips": 7943, "flagAt": 4046, "falseFlag": 0.048392, "power508": 0.407255, "power51": 0.548531, "power52": 0.971643, "power55": 1 }, { "flips": 8913, "flagAt": 4535, "falseFlag": 0.049225, "power508": 0.443614, "power51": 0.593243, "power52": 0.983216, "power55": 1 }, { "flips": 10000, "flagAt": 5083, "falseFlag": 0.049469, "power508": 0.48008, "power51": 0.636878, "power52": 0.990648, "power55": 1 }, { "flips": 11220, "flagAt": 5698, "falseFlag": 0.049253, "power508": 0.517041, "power51": 0.679575, "power52": 0.99515, "power55": 1 }, { "flips": 12589, "flagAt": 6388, "falseFlag": 0.048683, "power508": 0.554695, "power51": 0.721205, "power52": 0.997686, "power55": 1 }, { "flips": 14125, "flagAt": 7161, "falseFlag": 0.049556, "power508": 0.599671, "power51": 0.766689, "power52": 0.999053, "power55": 1 }, { "flips": 15849, "flagAt": 8029, "falseFlag": 0.049245, "power508": 0.641387, "power51": 0.806712, "power52": 0.999644, "power55": 1 }, { "flips": 17783, "flagAt": 9002, "falseFlag": 0.049495, "power508": 0.6858, "power51": 0.845543, "power52": 0.999886, "power55": 1 }, { "flips": 19953, "flagAt": 10094, "falseFlag": 0.048801, "power508": 0.726946, "power51": 0.878746, "power52": 0.999968, "power55": 1 }, { "flips": 22387, "flagAt": 11318, "falseFlag": 0.048708, "power508": 0.769309, "power51": 0.909093, "power52": 0.999993, "power55": 1 }, { "flips": 25119, "flagAt": 12691, "falseFlag": 0.049154, "power508": 0.811342, "power51": 0.93536, "power52": 0.999999, "power55": 1 }, { "flips": 28184, "flagAt": 14231, "falseFlag": 0.049473, "power508": 0.849956, "power51": 0.956174, "power52": 1, "power55": 1 }, { "flips": 31623, "flagAt": 15959, "falseFlag": 0.049136, "power508": 0.883393, "power51": 0.971518, "power52": 1, "power55": 1 }, { "flips": 35481, "flagAt": 17896, "falseFlag": 0.049907, "power508": 0.91438, "power51": 0.983075, "power52": 1, "power55": 1 }, { "flips": 39811, "flagAt": 20071, "falseFlag": 0.049072, "power508": 0.938059, "power51": 0.99028, "power52": 1, "power55": 1 }, { "flips": 44668, "flagAt": 22509, "falseFlag": 0.049338, "power508": 0.958224, "power51": 0.995003, "power52": 1, "power55": 1 }, { "flips": 50119, "flagAt": 25245, "falseFlag": 0.049193, "power508": 0.973161, "power51": 0.997637, "power52": 1, "power55": 1 }, { "flips": 56234, "flagAt": 28313, "falseFlag": 0.04959, "power508": 0.984046, "power51": 0.999014, "power52": 1, "power55": 1 }, { "flips": 63096, "flagAt": 31756, "falseFlag": 0.049252, "power508": 0.991036, "power51": 0.999627, "power52": 1, "power55": 1 }, { "flips": 70795, "flagAt": 35617, "falseFlag": 0.049865, "power508": 0.995489, "power51": 0.999881, "power52": 1, "power55": 1 }, { "flips": 79433, "flagAt": 39949, "falseFlag": 0.049847, "power508": 0.997904, "power51": 0.999967, "power52": 1, "power55": 1 }, { "flips": 89125, "flagAt": 44809, "falseFlag": 0.049673, "power508": 0.999123, "power51": 0.999992, "power52": 1, "power55": 1 }, { "flips": 100000, "flagAt": 50261, "falseFlag": 0.049723, "power508": 0.999678, "power51": 0.999999, "power52": 1, "power55": 1 }, { "flips": 112202, "flagAt": 56377, "falseFlag": 0.04999, "power508": 0.999898, "power51": 1, "power52": 1, "power55": 1 }, { "flips": 125893, "flagAt": 63239, "falseFlag": 0.049889, "power508": 0.999972, "power51": 1, "power52": 1, "power55": 1 }, { "flips": 141254, "flagAt": 70937, "falseFlag": 0.04978, "power508": 0.999994, "power51": 1, "power52": 1, "power55": 1 }, { "flips": 158489, "flagAt": 79573, "falseFlag": 0.049697, "power508": 0.999999, "power51": 1, "power52": 1, "power55": 1 }, { "flips": 177828, "flagAt": 89262, "falseFlag": 0.049665, "power508": 1, "power51": 1, "power52": 1, "power55": 1 }, { "flips": 199526, "flagAt": 100131, "falseFlag": 0.049937, "power508": 1, "power51": 1, "power52": 1, "power55": 1 }, { "flips": 223872, "flagAt": 112326, "falseFlag": 0.04984, "power508": 1, "power51": 1, "power52": 1, "power55": 1 }, { "flips": 251189, "flagAt": 126008, "falseFlag": 0.049667, "power508": 1, "power51": 1, "power52": 1, "power55": 1 }, { "flips": 281838, "flagAt": 141357, "falseFlag": 0.049656, "power508": 1, "power51": 1, "power52": 1, "power55": 1 }, { "flips": 316228, "flagAt": 158577, "falseFlag": 0.049994, "power508": 1, "power51": 1, "power52": 1, "power55": 1 }, { "flips": 354813, "flagAt": 177897, "falseFlag": 0.049961, "power508": 1, "power51": 1, "power52": 1, "power55": 1 }, { "flips": 398107, "flagAt": 199573, "falseFlag": 0.049973, "power508": 1, "power51": 1, "power52": 1, "power55": 1 }, { "flips": 446684, "flagAt": 223893, "falseFlag": 0.049743, "power508": 1, "power51": 1, "power52": 1, "power55": 1 }, { "flips": 501187, "flagAt": 251177, "falseFlag": 0.049777, "power508": 1, "power51": 1, "power52": 1, "power55": 1 }, { "flips": 562341, "flagAt": 281788, "falseFlag": 0.049927, "power508": 1, "power51": 1, "power52": 1, "power55": 1 }, { "flips": 630957, "flagAt": 316133, "falseFlag": 0.049812, "power508": 1, "power51": 1, "power52": 1, "power55": 1 }, { "flips": 707946, "flagAt": 354666, "falseFlag": 0.049874, "power508": 1, "power51": 1, "power52": 1, "power55": 1 }, { "flips": 794328, "flagAt": 397898, "falseFlag": 0.049882, "power508": 1, "power51": 1, "power52": 1, "power55": 1 }, { "flips": 891251, "flagAt": 446403, "falseFlag": 0.049874, "power508": 1, "power51": 1, "power52": 1, "power55": 1 }, { "flips": 1000000, "flagAt": 500823, "falseFlag": 0.049985, "power508": 1, "power51": 1, "power52": 1, "power55": 1 } ], "widget": { "flips": [ 10, 20, 50, 100, 200, 500, 1000, 2000, 5000, 10000 ], "sameSide": [ 0.505, 0.508, 0.51, 0.515, 0.52, 0.53, 0.55, 0.6 ], "flagAt": [ 9, 15, 32, 59, 113, 269, 527, 1038, 2559, 5083 ], "falseFlag": [ 0.010742, 0.020695, 0.032454, 0.044313, 0.038419, 0.048945, 0.046844, 0.046754, 0.048995, 0.049469 ], "power": [ { "sameSide": 0.505, "byFlips": [ 0.011652, 0.023015, 0.037929, 0.054501, 0.051769, 0.076135, 0.08691, 0.109355, 0.171676, 0.257843 ] }, { "sameSide": 0.508, "byFlips": [ 0.01223, 0.024508, 0.04156, 0.061451, 0.061403, 0.097247, 0.120942, 0.16812, 0.300395, 0.48008 ] }, { "sameSide": 0.51, "byFlips": [ 0.012628, 0.025548, 0.044134, 0.066457, 0.068568, 0.113534, 0.148292, 0.216899, 0.405021, 0.636878 ] }, { "sameSide": 0.515, "byFlips": [ 0.013672, 0.02831, 0.05113, 0.080349, 0.089291, 0.162486, 0.233468, 0.36868, 0.679752, 0.911578 ] }, { "sameSide": 0.52, "byFlips": [ 0.014789, 0.031315, 0.058983, 0.096335, 0.114354, 0.223458, 0.340519, 0.544661, 0.879937, 0.990648 ] }, { "sameSide": 0.53, "byFlips": [ 0.017258, 0.038116, 0.077501, 0.135098, 0.178624, 0.377232, 0.587977, 0.843282, 0.995219, 0.999993 ] }, { "sameSide": 0.55, "byFlips": [ 0.023257, 0.055334, 0.127345, 0.241491, 0.361997, 0.720803, 0.932231, 0.997483, 1, 1 ] }, { "sameSide": 0.6, "byFlips": [ 0.046357, 0.125599, 0.335613, 0.622533, 0.860336, 0.997863, 0.999999, 1, 1, 1 ] } ], "note": "One-sided exact test at 5%: flag the coin when the same-side count is at least flagAt[i] out of flips[i]. power[j].byFlips[i] = exact chance of a flag for a coin with that same-side chance; falseFlag[i] = exact chance a fair coin is flagged anyway (at most 5%, less because counts are whole numbers). A widget can apply the same rule to its own simulated flips and show the exact value beside them. Values printed as 1 are rounded (6 decimals): show them as >99.99%, never 100%." }, "formulas": { "power": "P(X >= flagAt | X ~ Bin(n, p))", "falseFlag": "P(X >= flagAt | X ~ Bin(n, 1/2)) <= 0.05", "approxSampleSize": "n = ((z_alpha × 0.5 + z_power × sqrt(p(1-p))) / (p - 0.5))^2, z = 1.645 (one-sided 5%) or 1.960 (two-sided 5%), 0.842 (80%) or 1.282 (90%)", "exactSampleSize": "staysAbove = the smallest n from which exact power stays at or above the target for every n scanned (to checkedTo); firstReached = the first n that touches it (power is a sawtooth in n)", "edgeInSds": "n(p - 1/2) / sqrt(n p (1-p)): the extra same-side landings measured in standard deviations of the count" } }, "study": { "note": "Recomputed from the counts Bartoš et al. report (178,079 same-side landings and 175,421 heads in 350,757 flips): plain binomial shares, 95% Wald intervals and z-scores against exactly 1/2. They match the paper's simple-binomial figures, 0.5077 [0.5060, 0.5094] and 0.5001 [0.4985, 0.5018]; the abstract's 0.508 [0.506, 0.509] is that same simple-binomial result rounded to three decimals, i.e. the share pooled over all flips. It is not 'the average person's bias': the paper's hierarchical model, which estimates the same-side chance for a typical flipper, gives 0.5098 [0.5050, 0.5147] (sources.bartos.figures.hierarchical). Print intervals with the *Display strings (fixed 4 decimals). Arithmetic on published counts, not a simulation.", "flips": 350757, "sameSideCount": 178079, "headsCount": 175421, "sameSideShare": 0.507699, "sameSideInterval95": [ 0.506, 0.5094 ], "sameSideDisplay": "0.5077 [0.5060, 0.5094]", "sameSideExtraOverHalf": 2700.5, "sameSideZ": 9.12, "headsShare": 0.500121, "headsInterval95": [ 0.4985, 0.5018 ], "headsZ": 0.14, "headsDisplay": "0.5001 [0.4985, 0.5018]", "averageFlipsPerPerson": 7307, "averageFlipsPerPersonNote": "A plain average (350,757 / 48). Contributions were very unequal (Table 1 includes people with 1,600 flips; 35 people produced 203,440 flips in on-site marathons), so write 'an average of about 7,300 flips each, with very different amounts per person', never 'each person flipped about 7,300 times'.", "standardErrorAt05": 0.0008442, "sdShareAt508": 0.0008441, "sdShareAt508Note": "exact SD of the same-side share over 350,757 flips of a coin that lands same-side exactly 50.8% of the time: the value the simulated replications' spread (results.studyReplications.sdShare) is compared with", "powerOfStudySize": { "flagAt": 175867, "power": 1, "powerDisplay": ">99.99%", "missChance": 2.467e-15, "missOneIn": "about 1 in 405 trillion", "note": "exact: the chance that a 350,757-flip experiment with a coin that lands same-side exactly 50.8% of the time is NOT flagged by the one-sided 5% test is missChance (a tail computed in double precision: only its first few digits are meaningful, hence missOneIn to 3 significant figures)" } }, "games": { "sameSide": 0.508, "sameSideFraction": "127/250", "bestOf": [ { "bestOf": 1, "pWin": 0.508, "pWinDisplay": "50.80%", "edgePoints": "0.80 points", "pWinIf051": 0.51, "pWinIf051Display": "51.00%", "expectedTosses": 1 }, { "bestOf": 3, "pWin": 0.511999, "pWinDisplay": "51.20%", "edgePoints": "1.20 points", "pWinIf051": 0.514998, "pWinIf051Display": "51.50%", "expectedTosses": 2.4999 }, { "bestOf": 5, "pWin": 0.514997, "pWinDisplay": "51.50%", "edgePoints": "1.50 points", "pWinIf051": 0.518745, "pWinIf051Display": "51.87%", "expectedTosses": 4.1246 }, { "bestOf": 7, "pWin": 0.517496, "pWinDisplay": "51.75%", "edgePoints": "1.75 points", "pWinIf051": 0.521866, "pWinIf051Display": "52.19%", "expectedTosses": 5.8117 } ], "formulas": { "bestOf": "P(win best of 2k+1) = sum_{j=k+1}^{2k+1} C(2k+1, j) p^j (1-p)^(2k+1-j), exact fractions with p = 127/250", "expectedTosses": "sum over the deciding toss t = k+1+j (j = the loser's wins) of t × C(t-1, j) × [p^(k+1)(1-p)^j + (1-p)^(k+1) p^j]" }, "callScenarios": [ { "id": "seesStartCallsIt", "who": "the caller can see which side is up and calls that side", "callerWins": 0.508, "callerWinsDisplay": "50.80%", "benefits": "caller", "edgePoints": "0.80 points", "simulated": "bestOf 1" }, { "id": "startHidden", "who": "the starting side is equally likely to be heads or tails (for example shaken in cupped hands) and hidden from the caller, so the call does not depend on it", "callerWins": 0.5, "callerWinsDisplay": "50.00%", "benefits": "nobody", "edgePoints": "0.00 points", "note": "Exactly 1/2 whatever the same-side bias, by symmetry: the caller's side is the starting side half the time. This needs the starting side to be equally likely to be either side (or the call to be a fair random choice) and independent of the call; hiding alone is not enough: a flipper who habitually starts heads up, out of sight, facing a caller who calls heads 70% of the time gives the caller 0.7 × 0.508 + 0.3 × 0.492 = 0.5048. The simulation draws the start with below(2).", "simulated": "callHidden" }, { "id": "flipperSetsStartAgainstCall", "who": "the call is made before the flip and the flipper, knowing it, starts the coin with the called side down", "callerWins": 0.492, "callerWinsDisplay": "49.20%", "benefits": "flipper", "edgePoints": "0.80 points", "simulated": "callBeforeFlipStartedAgainst" }, { "id": "slappedOnHand", "who": "caught, then turned over onto the back of the other hand (Diaconis, Holmes & Montgomery note this reverses the bias)", "landsOnStartingSide": 0.492, "landsOnStartingSideDisplay": "49.20%", "landsOnStartingSideIsDerived": true, "benefits": "whoever calls the side that started down", "note": "Our derivation, not a figure from either paper: Diaconis, Holmes & Montgomery say only that turning the caught coin over gives a bias opposite to the start, with no number. 0.492 = 1 - 0.508 assumes the caught coin lands same-side exactly 50.8% of the time and is then turned over once (the same arithmetic as flipperSetsStartAgainstCall). In prose, say the bias is reversed, or label 0.492 as derived from the 0.508 model. If the coin is sometimes turned over and sometimes not, they note that a significant amount of randomness may be added." } ], "overManyTosses": [ { "tosses": 100, "expectedWinsSeesStart": 50.8, "extraWinsOverFair": 0.8, "sdOfWins": 4.9994, "pWinMoreThanHalf": 0.524118, "pWinMoreThanHalfDisplay": "52.4%", "pWinMoreThanHalfFair": 0.460205, "pWinMoreThanHalfFairDisplay": "46.0%" }, { "tosses": 1000, "expectedWinsSeesStart": 508, "extraWinsOverFair": 8, "sdOfWins": 15.8094, "pWinMoreThanHalf": 0.682429, "pWinMoreThanHalfDisplay": "68.2%", "pWinMoreThanHalfFair": 0.487387, "pWinMoreThanHalfFairDisplay": "48.7%" } ], "scope": "All values assume a coin (or flipper) that lands on its starting side with exactly 0.508 on every toss, independently; real flippers in Bartoš et al. ranged from 0.487 to 0.601.", "note": "Who benefits is decided by who knows the starting side relative to the call: a caller who sees it and calls it (50.8%), a flipper who hears the call first and starts the coin the other way up (the caller drops to 49.2%), nobody when the starting side is hidden (exactly 50%). Best-of formats add only a little: the edge grows from 0.8 points to the values in bestOf. pWinIf051 is a 51% coin for comparison (exact); it is not a separate prediction of Diaconis, Holmes & Montgomery, whose quoted 0.51 is their mean estimate 0.508 rounded up (sources.dhm.meanEstimate)." }, "streaks": { "flips": 100, "repeatReal": 0.508, "repeatVirtual": 0.5, "model": "Each flip of a hand-flipped coin starts from the side the previous flip landed on (the procedure Bartoš et al. used), so it repeats the previous result with chance 0.508. The virtual coin repeats with chance exactly 1/2.", "condition": "The real-coin streak figures hold ONLY when every flip starts from the side the coin last landed on, as in the study, and the flipper has exactly a 50.8% same-side chance. In everyday tossing the flipper usually picks the starting side, and then the same-side bias need not make results repeat more often. Introduce any streak comparison with this condition.", "method": "exact: integer dynamic program over the current run length, weights 127/123 (real) and 1/1 (virtual) per flip after the first, divided by 250^99 or 2^99", "real": { "meanLongest": 7.1054, "sdLongest": 1.832, "pAtLeast": { "5": 0.977049, "6": 0.828601, "7": 0.572625, "8": 0.341012, "10": 0.097922 }, "pAtLeastDisplay": { "5": "97.70%", "6": "82.9%", "7": "57.3%", "8": "34.1%", "10": "9.79%" } }, "virtual": { "meanLongest": 6.9774, "sdLongest": 1.7926, "pAtLeast": { "5": 0.97169, "6": 0.806821, "7": 0.542337, "8": 0.314767, "10": 0.086659 }, "pAtLeastDisplay": { "5": "97.17%", "6": "80.7%", "7": "54.2%", "8": "31.5%", "10": "8.67%" } }, "difference": { "5": "0.54 points", "6": "2.18 points", "7": "3.03 points", "8": "2.62 points", "10": "1.13 points" }, "meanDifference": 0.128, "distribution": [ { "longest": 1, "real": 0, "virtual": 0 }, { "longest": 2, "real": 0, "virtual": 0 }, { "longest": 3, "real": 0.000195, "virtual": 0.000285 }, { "longest": 4, "real": 0.022755, "virtual": 0.028026 }, { "longest": 5, "real": 0.148449, "virtual": 0.164869 }, { "longest": 6, "real": 0.255975, "virtual": 0.264484 }, { "longest": 7, "real": 0.231614, "virtual": 0.22757 }, { "longest": 8, "real": 0.154271, "virtual": 0.146209 }, { "longest": 9, "real": 0.088818, "virtual": 0.081899 }, { "longest": 10, "real": 0.047678, "virtual": 0.042991 }, { "longest": 11, "real": 0.02474, "virtual": 0.021878 }, { "longest": 12, "real": 0.012627, "virtual": 0.010969 }, { "longest": 13, "real": 0.006393, "virtual": 0.00546 }, { "longest": 14, "real": 0.003224, "virtual": 0.002708 }, { "longest": 15, "real": 0.001622, "virtual": 0.001341 }, { "longest": 16, "real": 0.000815, "virtual": 0.000663 }, { "longest": 17, "real": 0.00041, "virtual": 0.000328 }, { "longest": 18, "real": 0.000206, "virtual": 0.000162 }, { "longest": 19, "real": 0.000103, "virtual": 0.00008 }, { "longest": 20, "real": 0.000052, "virtual": 0.00004 } ], "note": "pAtLeast[k] = chance that the longest run of identical results in 100 flips is k or more. For streak details on the site's wheels, the study why-random-feels-rigged has the full treatment; this only compares a restarted hand flip with the virtual coin." } }, "checks": [ { "name": "virtual coin (yes or no wheel, pickWeighted([1,1])): first entry (Yes / Heads), 20,000,000 flips", "kind": "share", "simulated": 0.500067, "exact": 0.5, "se": 0.000112, "z": 0.6, "pass": true }, { "name": "virtual coin: a flip repeats the one before it, 19,999,999 pairs", "kind": "share", "simulated": 0.500124, "exact": 0.5, "se": 0.000112, "z": 1.11, "pass": true }, { "name": "virtual coin: consecutive pairs HH/HT/TH/TT equally likely (Good's serial test, chi-square 2 df)", "kind": "chi-square", "simulated": 1.591681, "exact": 2, "se": 2, "z": -0.2, "pass": true }, { "name": "virtual coin: rejection step never fires on a two-entry wheel (redraws in 20,000,000 flips)", "kind": "sanity", "simulated": 0, "exact": 0, "se": 0, "z": 0, "pass": true }, { "name": "0.508 coin, 100 flips: the 5% test flags it (power)", "kind": "share", "simulated": 0.061065, "exact": 0.061451, "se": 0.000537, "z": -0.72, "pass": true }, { "name": "0.508 coin, 100 flips: more same-side than other-side landings", "kind": "share", "simulated": 0.524885, "exact": 0.524118, "se": 0.00112, "z": 0.69, "pass": true }, { "name": "virtual coin, 100 flips: the same test flags it anyway (false flag)", "kind": "share", "simulated": 0.04425, "exact": 0.044313, "se": 0.00046, "z": -0.14, "pass": true }, { "name": "0.508 coin, 1,000 flips: the 5% test flags it (power)", "kind": "share", "simulated": 0.121175, "exact": 0.120942, "se": 0.00163, "z": 0.14, "pass": true }, { "name": "0.508 coin, 1,000 flips: more same-side than other-side landings", "kind": "share", "simulated": 0.680625, "exact": 0.682429, "se": 0.00233, "z": -0.78, "pass": true }, { "name": "0.508 coin, 1,000 flips: mean same-side landings", "kind": "mean", "simulated": 508.0151, "exact": 508, "se": 0.079, "z": 0.19, "pass": true }, { "name": "virtual coin, 1,000 flips: the same test flags it anyway (false flag)", "kind": "share", "simulated": 0.047, "exact": 0.046844, "se": 0.00106, "z": 0.15, "pass": true }, { "name": "0.508 coin, 24,328 flips (exact 80% sample size): the 5% test flags it", "kind": "share", "simulated": 0.802, "exact": 0.801781, "se": 0.00728, "z": 0.03, "pass": true }, { "name": "100 simulated replications of 350,757 flips at 0.508: mean same-side share", "kind": "mean", "simulated": 0.50791, "exact": 0.508, "se": 0.0000844, "z": -1.06, "pass": true }, { "name": "100 simulated replications of 350,757 flips: spread (SD) of the same-side share", "kind": "sd", "simulated": 0.000882, "exact": 0.000844, "se": 0.00006, "z": 0.63, "pass": true }, { "name": "best of 1: the caller who sees the starting side and calls it wins the match", "kind": "share", "simulated": 0.50809, "exact": 0.508, "se": 0.0005, "z": 0.18, "pass": true }, { "name": "best of 3: the caller who sees the starting side and calls it wins the match", "kind": "share", "simulated": 0.511273, "exact": 0.511999, "se": 0.0005, "z": -1.45, "pass": true }, { "name": "best of 5: the caller who sees the starting side and calls it wins the match", "kind": "share", "simulated": 0.515068, "exact": 0.514997, "se": 0.0005, "z": 0.14, "pass": true }, { "name": "best of 7: the caller who sees the starting side and calls it wins the match", "kind": "share", "simulated": 0.517821, "exact": 0.517496, "se": 0.0005, "z": 0.65, "pass": true }, { "name": "call in the air, starting side hidden (caller always calls heads): caller wins", "kind": "share", "simulated": 0.500302, "exact": 0.5, "se": 0.000354, "z": 0.85, "pass": true }, { "name": "call made before the flip, flipper starts the coin with the other side up: caller wins", "kind": "share", "simulated": 0.492038, "exact": 0.492, "se": 0.000354, "z": 0.11, "pass": true }, { "name": "longest run in 100 flips, hand flip restarted from where it landed (0.508): mean", "kind": "mean", "simulated": 7.100805, "exact": 7.105414, "se": 0.0041, "z": -1.13, "pass": true }, { "name": "longest run in 100 flips, hand flip restarted from where it landed (0.508): 7 or more in a row", "kind": "share", "simulated": 0.572545, "exact": 0.572625, "se": 0.00111, "z": -0.07, "pass": true }, { "name": "longest run in 100 flips, hand flip restarted from where it landed (0.508): 8 or more in a row", "kind": "share", "simulated": 0.33929, "exact": 0.341012, "se": 0.00106, "z": -1.62, "pass": true }, { "name": "longest run in 100 flips, virtual coin: mean", "kind": "mean", "simulated": 6.975765, "exact": 6.977432, "se": 0.00401, "z": -0.42, "pass": true }, { "name": "longest run in 100 flips, virtual coin: 7 or more in a row", "kind": "share", "simulated": 0.54109, "exact": 0.542337, "se": 0.00111, "z": -1.12, "pass": true }, { "name": "longest run in 100 flips, virtual coin: 8 or more in a row", "kind": "share", "simulated": 0.31429, "exact": 0.314767, "se": 0.00104, "z": -0.46, "pass": true } ] }