Secret Santa Draws: How Often the Hat Goes Wrong, and a Fair Fix

The short answer

• A plain hat fails more often than it works. With 6 or more people, someone draws their own name in 63.2% of draws (62.5% with 4). You need 2.72 full draws on average, counting the one that works, and 25.3% of the time it takes more than three.
• "Put your own name back and draw again" is not fair. With 10 people it dead-ends 7.56% of the time, when the last person is left holding only their own name. When it does finish, the last person to draw gets the person who drew just before them 18.0% of the time, against a fair 11.1%.
• Two fair ways to draw. Redraw the whole hat whenever anyone gets themselves, or use a loop: shuffle the list once, and each person gives to the name below theirs, the last to the first. The loop never gives anyone themselves or a swapped pair, and each person is equally likely to get each other person.
• Couples make the hat much worse. With 10 people including 3 couples who must not draw each other, only 18.5% of hat draws are valid (5.40 full draws on average). A shuffled loop is valid 48.4% of the time.

The exact values come from exact counting formulas and, for the put-back rule, a full count of every draw for up to 10 people. Another 97.2 million simulated random draws through random.js, the file behind every pick on Wheel Cue, agree with them. Percentages are exact unless the sentence says "simulated". The helper further down draws a valid Secret Santa for your own list.

How often a hat draw goes wrong

In a hat draw for Secret Santa, or any name-draw gift exchange, everyone writes their name on a slip, the slips are mixed, and each person takes one. If the hat is well mixed, every way of handing out the names is equally likely. The draw works only if nobody takes their own slip, which mathematicians call a derangement.

For 10 people there are 3,628,800 ways to hand out the names, and 1,334,961 of them leave nobody with their own name: 36.8%. That share settles almost at once on 1/e, about 36.8% (e is about 2.71828), so group size hardly matters:

In 1,000,000 simulated hats of 10 people, nobody drew their own name in 36.7%.

With 10 people, exactly one person gets themselves 36.8% of the time and two or more 26.4%, so the most common failure is one person quietly holding their own name.

If the rule is to redraw the whole hat until nobody has themselves, expect 2.72 full draws on average, the first included. That is slow, but fair: each redraw is a fresh hat, so the draw you keep is equally likely to be any valid one.

"Put it back and draw again" is not fair

A common shortcut is quicker: whoever draws their own name puts it back and draws again. We modeled it as people do it, drawing in a fixed order, and found two problems.

It can dead-end

When the last person's turn comes, only one slip is left. If it is their own, there is nothing to swap it with and the draw has to start again. That happens in 7.56% of draws with 10 people (1 in 13), and between 4.20% (20 people) and 13.9% (4) across all sizes in the table above. Nobody finds out until the end, after everyone has seen their slip.

When it finishes, the pairings are tilted

The bigger problem is quieter. Number people in the order they draw. Under the put-back rule, the last person to draw ends up with the person just before them far more often than with anyone else:

With 10 people, the last to draw gets the second-to-last 18.0% of the time, 1.62 times the fair 11.1%, and gets the first person to draw only 9.17%. In 1,000,000 simulated put-back draws, the last person got the second-to-last in 17.9% of those that finished.

The reason is the put-back itself. You can never take your own slip, so after your turn your name is more likely than most to still be in the hat, and the people who draw right after you pick it up. The second-to-last person is the extreme case: if their own slip is one of the last two, they must take the other one, and the last person gets theirs.

Relative to a fair draw, the last person's tilt grows with the group: with 20 people they get the second-to-last 9.20% of the time, 1.75 times the fair 5.26%. Looked at as complete draws, the most likely one comes up exactly n − 1 times as often as the least likely for groups of 4 to 10 (9 times with 10 people).

For a casual exchange the tilt is modest: nobody is shut out, the last person's chances range from 9.17% to 18.0% against a fair 11.1%, and some pair draws each other in 39.2% of put-back draws against 39.3% in a fair one. The dead end, which forces a restart after everyone has looked, is reason enough to switch.

What a fair draw looks like, and why a loop is simpler

Suppose every valid draw is equally likely, as the redraw-the-whole-hat rule gives. Then each giver is equally likely to get each of the other people: 1/9 each with 10 people. But the draw as a whole often has features groups notice:

• Two people draw each other. With 10 people, some pair swaps gifts in 39.3% of fair draws, and it stays between 38.5% and 39.6% from 6 people up. Any two named people draw each other 1.11% of the time.
• The gifts form separate circles. Follow the gifts from person to person and you come back to the start. With 10 people, one single loop through everyone happens only 27.2% of the time; two separate loops 46.7%, three 22.7%. A fair draw has 2.02 loops on average.

Neither is unfair, but both matter if gifts are opened in a chain or a swapped pair would spoil the surprise.

The loop method

Shuffle the list of names once, then read it as a circle: each person gives to the name below theirs, and the last name gives to the first. The random name picker does the shuffling with its Shuffle button, the same unbiased shuffle the site uses everywhere.

• Nobody can draw themselves, and no two people can draw each other once there are 3 or more names.
• Everyone is in one loop, so a gift chain never stalls.
• Each person gets each other person with chance 1/(n − 1), the same as in a fair hat, and every one of the (n − 1)! possible loops is equally likely.

The loop is a smaller menu: of the 1,334,961 valid draws for 10 people, 362,880 are single loops (27.2%). It never produces a swap, so nobody buys for the person buying for them. The flip side: anyone who knows a loop was used can rule out their own recipient as their Santa. In 1,000,000 simulated shuffles of 10 names, person 1 gave to each of the other 9 between 11.1% and 11.2% of the time.

There is one trade-off: a hat keeps the pairings secret from everyone, but whoever shuffles the list sees every pairing.

Couples and last year's matches

Some groups add a second rule: partners must not draw each other, or nobody gets the same person as last year. Each extra rule makes a plain hat fail more often:

With 10 people and 3 couples, a plain hat is valid only 18.5% of the time, so redrawing until both rules hold takes 5.40 full draws on average. And 49.6% of the hats in which nobody drew themselves still hand someone their own partner, so checking only for self-draws is not enough.

The loop holds up better: shuffle until no couple sits next to each other. With 3 couples among 10 that takes 2.07 shuffles on average. A group of 4 couples needs 3.39 shuffles against 8.48 full hat draws.

"Not the same as last year" is about as strict as a group made up entirely of couples, since each person again loses one possible recipient. If last year's draw was one loop, a plain hat avoids every repeat only 12.1% of the time with 10 people (12.1% for 5 couples), and that share tends to 13.5% in large groups. A new loop is valid 33.1% of the time, 3.02 shuffles on average.

Rules shift the odds a little

Shuffling until every rule holds, as the helper below does, makes every draw that follows the rules equally likely. But a rule changes who can get whom, so the chances are no longer all 1/(n − 1). With 10 people and 3 couples, against 1/9 (11.1%) with no couples:

• someone without a partner in the group draws a particular other such person 10.1% of the time;
• someone without a partner draws a particular member of a couple, or the other way round, 11.6% of the time;
• someone in a couple draws a particular member of another couple 13.4% of the time.

Each person in a couple has one fewer possible Santa, so the others draw them a little more often: the rule at work, not a flaw in the draw.

A fair Secret Santa with the name picker, step by step

Which method to use

• Everyone in one room, no extra rules: a paper hat with the redraw rule below. Nobody, not even the organizer, sees the pairings.
• Couples, last year's matches, or a group that is not in one place: the helper above, run by someone outside the group or an organizer everyone trusts to know.
• Just a quick list: the random name picker, whose Shuffle button does the loop method. You do not spin the wheel at all:

1. Press Clear under the Names box to remove the sample names, then type or paste your group, one name per line. If two people share a name, tell them apart, such as "Sam R." and "Sam T.", or they become two identical lines.
2. Decide the extra rules before you shuffle, such as couples or last year's matches, and write them down.
3. Press Shuffle once.
4. Read the list as a circle: each name gives to the name below it, and the last name gives to the first.
5. If you set a rule, check it: for couples, the two names must not sit next to each other, and the first and last names count as neighbors; for last year's matches, nobody's name may sit directly above the person they gave to last year. If the list breaks a rule, press Shuffle again and keep the first list that passes.
6. Copy the pairings into a note and send each giver their own person privately.

Two habits keep this fair. Never shuffle again because you dislike a pairing that follows the rules; that is choosing, not drawing. And do not send the shuffled wheel itself: a Send this wheel link carries the whole list in its order, so anyone who opens it can read every pairing.

Names typed into a wheel are not stored by the site, and Save keeps a wheel only in your own browser, so skip it on a shared computer.

If you would rather use a paper hat

Use the redraw rule: if anyone draws their own slip, everyone puts their slip back and the whole hat is drawn again. Announce it first, and expect 2.72 full draws on average. Each person checks their slip privately and says only "mine is me" (or "mine is my partner", if couples are kept apart).

What to avoid

• Spinning the wheel in front of the group. One spin per person with Remove & spin gives the same chances as a paper hat (the last person takes the name left), but it is not secret: every name shows on screen and in the Picked names panel.
• Fixing a draw by hand. Swapping two people to undo a match can make some pairings more likely than others. Redraw instead.

Pairing people every week instead? See random partners week after week.

How to check these numbers

The script and its raw results are linked at the top of the page. Save the script and random.js in one folder and run node secret-santa-draw-odds.mjs random.js results.json with Node.js 18 or later. It took about 130 seconds on our PC. The exact values will match this page; the simulated ones will differ in the last decimals.

How the script works:

• The hat is simulated on the name picker, one spin per person over the names still on the wheel, the last taking the last name, which gives exactly the chances of a well-mixed paper hat. The put-back rule re-spins when a person gets their own name and restarts after a dead end; other fixes, such as swapping with a neighbor, are not modeled. The loop is one call to CueRandom.shuffle, as the Shuffle button makes. The 97,173,556 simulated random draws count one per spin and n − 1 per shuffle of n names.
• Exact values use whole-number arithmetic: derangement counts for the hat, inclusion–exclusion for couples, ménage numbers for last year's loop, a dynamic program and a full enumeration of every draw for the put-back rule (4 to 10 people), and an exact step-by-step calculation for larger groups. Brute-force counts of every arrangement for up to 8 people cross-check the formulas.
• Checks. Each of the 64 simulated-vs-exact checks allows four standard errors; the largest gap in this run was 2.88 standard errors. 10 chi-square tests check that draws meant to be equally likely are, and that the put-back draws match their exact unequal chances. 3 sanity checks confirm every shuffled list forms one loop with no self-draws and no mutual pairs. All 77 passed. With this many checks, a rerun may very rarely fail one by chance.

The shuffle and the wheel are tested at scale in the fairness report, and why random feels rigged explains why fair draws often look suspicious.