# Homework Help: Finding Conditonal Probability

1. Oct 13, 2009

### mckallin

The problem statement, all variables and given/known data
There are w white balls and b black balls in a bowl. Randomly select a ball from the bowl and then return it to the bowl along with n additional balls of the same color. Another single ball is randomly selected from the bowl(now containing w+b+n balls) and it is black. Show that the conditional probability that the first ball selected was white is w/(w+b+n)

Relevant equations
The conditional probability of an event A, given that an event B has occurred, is equal to:
P(AlB)=P(AnB)/P(B)

The attempt at a solution
This is my last question of my assignment and I can't figure out even how to get the first step. The condition we know here is an event happened afterward, so I am even confused if I should use the equation above. I try to list the sample points which are A(1st-w, 2nd-w), B(1st-w, 2nd-b), C(1st-b, 2nd-b), D(1st-b, 2nd-w), and the possible sample points should be B or D. Then I try the conditional probability for B:

Sample point B :

Being the first selecting:
P(w)=w/(w+b) P(b)=b/(w+b)

By sample point B, it supports that the first selected ball is white, second is black, so using the equation:
P(blw)=P(bnw)/P(w)=P(bnw)/[w/(w+b)]=b/(w+n+b)
so, P(bnw)=[w/(w+b)]*[b/(w+n+b)]

Here it already looks strange because B is just one of the sample points, but Ｉ still continue:

P(wlb)=P(wnb)/P(b)={[w/(w+b)]*[b/(w+n+b)]}/[b/(w+b)]=w/(w+b+n)

I got the answer, but I have no feeling for that. It was that I was just putting something into an equation with no reason. But when I tried other ways, it even went worse. If anyone can give me some ideas just like how I should start to prove and that will be great.

2. Oct 13, 2009

### PAR

I don't understand what the question is asking: "Show that the conditional probability that the first ball selected was white is w/(w+b+n)" This doesn't make any sense. The probability that the first ball selected is white is clearly w/(w+b) unless i am horribly, horribly mistaken. Reread the question.

EDIT: Oh never mind, I missed the part where the second ball selected was black.

After studying this problem further, I decided I don't know enough to help you out, sorry. Hopefully someone else can explain.

Last edited: Oct 13, 2009
3. Oct 13, 2009

### mckallin

I think it means that at first the color of the ball is unknown, and given the condition that the second selected ball is black; thus, under this condition what the probability of the the first ball selected was white is.
That is what I got. I agree with your meaning though, but anyway that is what the question is asking, driving me cruzy.