Secret Santa Draws: There Is a 63% Chance Someone Picks Themselves
If your group draws names from a hat for Secret Santa, there is about a 63% chance that at least one person pulls out their own name, whether the group has 5 people or 50. Only about 37% of draws work first time. This famous result is called the derangement problem, and the answer is linked to the number e.
Key facts
- A random draw works (nobody gets themselves) only about 36.8% of the time.
- The chance hardly changes with group size: 37.5% with 4 people, 36.8% with 12, 36.7% with 20.
- On average you need about 2.7 attempts to get a valid draw.
- A wheel with elimination mode lets each person spin while their own name is removed first.
The numbers
| Group size | Draw works first time |
|---|---|
| 3 | 33.2% |
| 4 | 37.5% |
| 5 | 36.6% |
| 6 | 36.9% |
| 8 | 36.7% |
| 10 | 37.0% |
| 12 | 36.8% |
| 20 | 36.7% |
| 30 | 36.9% |
These figures come from 200,000 simulated draws per group size. The exact value for larger groups is 1/e, or 36.79%, where e is the mathematical constant 2.718…
Why it does not get better with bigger groups
In a bigger group, each person is less likely to draw themselves (1 in 20 instead of 1 in 5), but there are more people who could. Those two effects cancel out almost perfectly, so the chance stays near 37%.
A fairer way to draw
1. Put everyone’s name on the random name picker.
2. For each person in turn, remove their own name, spin, write down who they got, then put their name back and remove the person they drew.
3. Or simply redraw whenever someone gets themselves: you will need about 2.7 tries on average.
Sources
- Our own simulation: 200,000 random draws per group size, secure random number generator.
- Background: Derangement, Wikipedia.
