# Finding the probability of drawing two white balls given that both balls are of the same color.

This is a problem of conditional probability:

There are two boxes X and Y. X contains 5 balls: 2 are blue, 2 white and 1 is gray. Y contains 4 balls: 2 are blue, 1 is white and 1 is gray.

I have to calculate the probability of both balls being white given that both are of the same colour. One ball is going to be taken from each box.

This problem, at least when I look to the formula of conditional probability ((P(A|B) = P(A and B) / P(B)), isn't very intuitive to me.

In this case, shouldn't the probability be P(White and White | Same Colour) = P[(White and White) and Same Colour)/P(Same Colour)?

My text book says it's going to be P(White and White) / P(Same Colour).

Could you guys help? Thank you very much and sorry about the wall text.

Edit: missed a sentence.

• I think you're missing a sentence. Is one ball drawn from each box? Nov 2 '17 at 13:01
• Yes, that is relevant information that I have forgotten. I am going to edit that, thank you. Nov 2 '17 at 13:13

You are both right. What you haven't yet noticed is that if the balls are both white, then they necessarily have the same colour. So we have $$P((\text{white and white})\text{ and same colour}) = P(\text{white and white})$$ Thus you have used the formula directly, while your text book has used the formula then applied a small simplification. Note that this simplification, while small, is a relatively crucial step towards being able to calculate anything.