What's the probability that the ball (taken from the first box and dropped in the second box) was white?

Question

I have one pretty tricky question:

There are two boxes. Each one has 7 balls: 3 white and 4 black. One ball was taken from the first box and dropped in the second one. After that, ball was taken from the second box and it was black. What's the probability that the ball, that was taken from the first box and dropped in the second one, was white?

I've tried solving it with combinatorics, and got the answer: 3/7 but it was not the right one.

Any help appreciate, Thanks!

Answer 1

Yes, your calculation is somewhat off.

Tip: Evaluate the following

• $P(W_1)\qquad$ The probability of adding a white to the second jar.
• $P(B_2\mid W_1)$ The probability of extracting a black given that a white had been added to the original 7.
• $P(W_1^\complement)\qquad$ The probability of adding a black to the second jar,
• $P(B_2\mid W_1^\complement)$ The probability of extracting a black given that a black had been added to the original 7.

Then use Bayes' Rule: $$\mathsf P(W_1\mid B_2)=\dfrac{\mathsf P(W_1)\mathsf P(B_2\mid W_1)}{\mathsf P(W_1)\mathsf P(B_2\mid W_1)+\mathsf P(W_1^\complement)\mathsf P(B_2\mid W_1^\complement)}$$