Wikipedia:Reference desk/Archives/Mathematics/2023 September 8

= September 8 =

Probability question
It seems to me the following problem should be really easy to work out, but I'll be damned if I can. You have a bag of infinite balls of 5 different colours, of which you draw 35. How do I work out the probability of drawing a particular number (between 0 and 35) of a particular colour of ball in these 35 draws? (It's for a game I play, I'm trying to work out if the boards are generally fair, or stacked). Many thanks BbBrock (talk) 15:23, 8 September 2023 (UTC)


 * We need some assumptions. The first is that the colours are present in the same ratios, so that (for a single draw) drawing a red ball is just as likely as drawing a blue ball, or a green ball (if these are three of the five colours). Also, the balls are supposed to be drawn independently, so at each subsequent draw each of the 5 possible outcomes is equally likely, regardless of what went on before. An equivalent problem is that there are just 5 balls in the bag, and that after each draw the drawn ball is returned. So what is the probability that (say) 9 of the 35 draws come up with a red ball? Generalizing this, let $$n$$ be the number of draws, and $$X$$ the number of observed occurrences of a specific outcome. (So in the example $$n=35$$ and $$X=9.$$) Also, let $$p$$ denote the probability of the specific outcome for a single draw. (In the example, $$p=\tfrac 15=0.2.$$) Then the probabilities that $$X$$ assumes a specific value $$k,$$ where $$k$$ ranges from $$0$$ to $$n,$$ are given by the so-called binomial distribution. In a formula:
 * $$\Pr(X=k)=\binom{n}{k}p^k(1-p)^{n-k}=\frac{n!}{k!(n-k)!}p^k(1-p)^{n-k}.$$
 * So for the example, $$\Pr(X=9) = \binom{35}{9}0.2^90.8^{26}\approx 0.10926.$$ --Lambiam 15:58, 8 September 2023 (UTC)


 * Many thanks Lambian. My schooldays are a long way behind me, and I'd forgotten the expression "binomial distribution". Thanks again! BbBrock (talk) 16:47, 8 September 2023 (UTC)


 * To add a bit, on Lambiam's assumptions the number of colours has relevance only to the denominator of the value of the probability of a particulat one in a single draw. For your case, 9 red (say) out of 35 means 24 not-red, hence only two possible outcomes on each draw, hence binomial. 2A00:23C6:AA0D:F501:DF0:92C9:5A41:EB60 (talk) 14:08, 11 September 2023 (UTC)