Consider an urn containing 12 balls, of which 8 are white. A sample of size 4 is to be drawn with replacement (without replacement). What is the conditional probability (in each case) that the first and third balls drawn will be white given that the sample drawn contains exactly 3 white balls?
So I'm having some trouble with the version of this problem that requires replacement.
Let, E be the event that the first and third draws are both white balls, and F be the event that the sample contains exactly 3 white balls.
So I'm looking for the conditional probability of E given F which is;
Where the intersection is the probability of drawing 1 non-white ball and 3 white balls. And since there are two valid permutations (wwwn,wnww) we introduce a factor of two
My problem is with the probability of F; which I'm defining as the odds of drawing one non-white ball and 3 white balls. I'm getting an answer that is less then the intersection of E and F so it's clearly nonsense.
Do I need to permute the balls in F or something?