Probability answers with receipts

Odds you can check yourself

Each answer here comes from a large simulation, published in chunks. Your browser can replay any chunk you pick and confirm it matches the published one, game for game. Where an exact answer exists, your browser works it out too.

The runs grow every night. While the Mac that runs them would otherwise sleep, it adds billions of games, and a second program written in a different language replays random chunks of every new batch before it is published. Most recent night (2026-10-31): 12,545,000,000 games added.

86questions
15.05 billionsimulated games
126,000replayable chunks
12.1CPU-hours

Secret Santa: does anyone draw their own name?

Everyone puts their name in a hat and draws one, with no do-overs. How often does at least one person end up with their own name? Surprisingly, group size barely matters: from about six people on, it sits near 63.2%. Every size here has an exact answer, from counting the shuffles where nobody gets their own name.

24 more questions

Secret Santa from a hat: does the draw get stuck?

The usual fix is to put your own name back and draw again. That works until the last person reaches the hat and finds only their own name left. With the couples rule, partners can't draw each other either, which makes it worse. Exact answers exist up to 14 people; for bigger groups the simulation stands on its own.

32 more questions

Dice: rolling all of a kind

Roll a handful of six-sided dice, keep every die showing your most common face, and re-roll the rest, up to three rolls in all. This is the keep-your-best strategy most players use when chasing five of a kind. The exact answers come from a Markov chain over every possible roll.

8 more questions

Collecting: how many packs to finish the set?

Every pack holds a few different items from the set, all equally common, and you never trade or swap. How many packs until you own every item? The average is the famous coupon-collector number, but the spread is what matters when you're buying: how many packs cover half, 90% or 99% of collectors.

6 more questions