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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, $p=\frac{1}{5}=0.2.$) 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:

$\mathrm{Pr}(X=k)=(\genfrac{}{}{0ex}{}{n}{k})p^k(1-p{)}^{n-k}=\frac{n!}{k!(n-k)!}{p}^k(1-p{)}^{n-k}.$

So for the example, $\mathrm{Pr}(X=9)=(\genfrac{}{}{0ex}{}{35}{9})0.2^{9}0.8^{26}\approx 0.10926.$  --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)

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