# Black and white balls in 2 boxes and probability that I pick the white ball.

Maybe there already is solution for that and if it is so, then maybe someone can tell me where I can find it.

I have 2 boxes. In first box there are 3 white and 2 black balls. In second box there are 4 black and 4 white balls. Then I randomly pick one ball from first box and put it in second box. What is the probability that I pick white ball from the second box?

I think that I need to calculate combinations for black balls and then for white balls (for second box after I put in random ball from first box) and then multiply them.

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Why not condition on which ball will be picked, and then combine the cases with the correct weight? –  gt6989b Mar 6 '13 at 18:19
Both answers give way too much detail, I like gt6989b´s comment. –  Adam Nov 13 '13 at 11:48

Case 1

You picked a white ball from the first box. This happens with a chance of $3/5$. Putting it into the second box, now there are $5$ white and $4$ black balls. Now the chance to pick a white ball is $5/9$. So the total chance in case 1 is $3/5 \cdot 5/9 = 1/3$.

Case 2

You picked a black ball from the first box. This happens with a chance of $2/5$. Putting it into the second box, now there are $4$ white and $5$ black balls. Now the chance to pick a white ball is $4/9$. So The total chance in case 2 is $2/5 \cdot 4/9 = 8/45$.

Combining the two cases, the chance is $$1/3 + 8/45 = 23/45 \approx 51\%.$$

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HINT: You have to look at the two possible cases, what is the probability of taking a white ball of the first box (3/5) and taking a white ball of the second box (there are now 5 balls in the second box, so 5/9), than look at the second case, where you take a black ball of the first box.

Combine these two cases with addition.

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