Secret Santa with 16 people: what is the chance someone draws their own name?
16 people each draw one name from a hat, with no redraws. How often does at least one person get their own name?
simulated: 63.2125% ± 0.0065% (95%), 132,746,173 of 210,000,000 games
In a Secret Santa with 16 people and no redraws, the chance that someone draws their own name is 63.212%. Worked out exactly and confirmed by 210,000,000 simulated games that you can replay on this page.
What the computer did
- Games
- 210,000,000 (2,000 chunks of 10,000 to 200,000, in 2 segments)
- Compute
- 0.10 CPU-hours; 0.01 hours on arm64, 16 cores, Darwin 25.5.0
- Engines
- JavaScriptCore (jsc), Python 3.14.8; every segment is spot-checked by replaying chunks in the other language
- Ran
- 2026-10-31T14:46:19Z to 2026-10-31T16:39:30Z
- Seed
- 602326568 (from this page's address); chunk k uses xoshiro128** seeded by splitmix32(seed ⊕ k·0x9E3779B1)
- Exact method
- derangement count: D(n)/n! is the chance nobody draws themselves
- Code
- sims/rng.py sha256 b81126f5f114cdf1…
sims/kinds/selfdraw.py sha256 d0ab9d3be0a3430b…
site_src/sims.js sha256 5cd91ae0e3f8246e…
Earlier segments: see the segment list below.
All 2 segments of this run
Three checks your browser runs
The totals add up. Adding up every published chunk…
It agrees with the exact answer. Your browser is working out the exact answer by a separate method…
The chunks are real. Pick chunks below and your browser replays them from the published seed. Each must match to the last game.
Replay any chunk
The run is split into 2,000 chunks of 10,000 to 200,000 games. Each chunk has its own random stream, built from the seed and the chunk number, so any one of them can be replayed alone. Your browser picks which ones at random, so the run could not have been prepared for your choice.
No chunks replayed yet. Each cell above is a chunk; replayed ones turn solid.
| Chunk | Published | Your browser | All counts | Games | ms |
|---|
What your replays rule out
Suppose someone had made up a share of the chunks instead of running them. These are the chances your replays so far would have landed on at least one made-up chunk.
Breakdown against the exact answer
| Outcome | Simulated | Exact | Off by (std. errors) |
|---|---|---|---|
| At least one person draws their own name | 63.2125% | 63.2121% | +0.12 |
| Two or more people draw their own name | 26.4229% | 26.4241% | -0.40 |
Independent simulations by local models
Each model below was given only the rules of this family of questions, in plain English, and asked to write its own simulation. It never saw our code. Its program was run in a sandbox on two test cases and compared with the exact answers.
- gemma4:26b agrees z = -0.31; z = -1.11
- gpt-oss:20b agrees z = +1.22; z = -1.29
- qwen2.5-coder:7b disagrees z = -586.22; z = -538.55
- qwen3.6:27b-mlx agrees z = +1.22; z = -1.29
What a match proves, and what it doesn't
- A matching chunk proves those games come from the published code and seed. Nobody could guess every count exactly without playing the games.
- It does not prove the code plays the game described. That is what the exact answer and the independent simulations are for.
- The run (Python) and the replay (JavaScript) were written by the same author, so they are two copies of one program. Their agreement shows the numbers were computed, not invented.
- One reader replaying a few chunks rules out large fakes, not small ones. Every reader's replays are added up, so together they cover the ledger.
Related questions
- Secret Santa with 15 people: what is the chance someone draws their own name?
- Secret Santa with 17 people: what is the chance someone draws their own name?
- Secret Santa with 14 people: what is the chance someone draws their own name?
- Secret Santa with 18 people: what is the chance someone draws their own name?
- Secret Santa with 13 people: what is the chance someone draws their own name?
- Secret Santa with 19 people: what is the chance someone draws their own name?