WORKSHOPA KNOWN BAG

What does
this chance mean?

A mathematical illustration of fictional board-game tokens, with no money, stakes or rewards.

Whole tokens only: 0–20 of each type and 1–6 draws. The bag must contain at least as many tokens as you draw.

AT LEAST ONE TARGET

53.3%

24 of 45 equally likely unordered samples

Two tokens are drawn without putting the first one back.

Count the samples

For N total tokens and d draws, there are C(N,d) equally likely unordered samples. If O tokens are not targets, C(O,d) samples contain no target. The chance of at least one target is 1 − C(O,d) / C(N,d). A combination count is zero when there are too few tokens to fill that sample.

Check the default example

With three targets and seven other tokens, drawing two gives 45 possible pairs. Of these, 21 contain only other tokens, leaving 24 pairs with at least one target. The displayed chance is therefore 24/45, rounded to 53.3%.

Keep the assumptions visible

Tokens are drawn uniformly and are not replaced during the draw. The composition is known. If a game breaks any of these assumptions, this particular calculation does not describe it. A percentage never guarantees a single result.