Calculating Probabilities for Multiple Colors of M&M's from a Bag

AI Thread Summary
To calculate the probability of drawing exactly 2 blue and 3 brown M&M's from a bag containing 22.5% blue and 12.5% brown, a multivariate hypergeometric distribution should be used due to the lack of replacement. The binomial distribution is inappropriate in this case since the draws are not independent. If an infinite number of M&M's were assumed, a multinomial distribution could be applied instead. The discussion highlights the importance of choosing the correct statistical model based on the sampling method. Understanding these distributions is crucial for accurate probability calculations in scenarios involving dependent events.
OneSquared
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I have a bag of M&M's that is 22.5% Blue, 12.5%Brown, and 65% other.

If I pull 12 M&M's from the bag, what is the probabiliity that exactly 2 are blue and 3 are brown?

I used the binomial to find the probability of 2 blue and 3 brown, and I want to multiply them together to get the answer, but wouldn't that assume that the two are independent? Obviously they are not, because any time I pull out a blue M&M, it is one time I have not pulled out a brown M&M.

Can someone shed light on how to solve this? Thanks!
 
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There is no replacement, so you can't use a multinomial distribution. You want a multivariate hypergeometric distribution. I found a formula on the web for it here:

http://www.agner.org/random/distrib.pdf
 
Correction - if you assume an infinite number of M&M's in the bag, then it's probably safe to use the multinomial distribution. The formula for that is also in the link I provided.
 
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