# 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.

• 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

## 3 Answers

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 $$1/3 + 8/45 = 23/45 \approx 51\%.$$

We know there are $9$ balls in the second box when you draw from it. Of these, $4+\frac25$ are expected to be black and $4+\frac35$ are expected to be white. Since the denominator is known and only the numerator is random, we can apply linearity of expectation and find the probability of picking a white ball as the expected number of white balls picked:

$$\frac{4+\frac35}9=\frac{23}{45}\;.$$

• Applying the linearity of expectation results in a wonderfully clear solution! – buruzaemon May 19 '16 at 3:33

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.