Hi, another actuary problem here:

Two bowls each contain 5 black and 5 white balls. A ball is chosen at random from bowl 1 and put into bowl 2. A ball is then chosen at random from bowl 2 and put into bowl 1. Find the probability that bowl 1 still has 5 black and 5 white balls.

So the actuary answer had more algebra, I will give their solution if I am wrong, but it seems I am right. I thought of the problem like this:

We denote P[A] and as the white ball transferring from bowl 1 to bowl 2. So then we denote P[A'] as black ball transferring from bowl 1 to bowl 2.

Also we denote P[B] as the white ball going back to bowl 1. We then denote P[B'] As the black ball going back to bowl 1.

P[ B | A]= 6/11 <--- this is the probability of white ball going back to bowl 1, given that the white bowl went into bowl 2.

Similiarily, we then know that P [B' | A'] = 6/11 as well, because it is the same event happening just with different colored ball.

Then we also know that P[ B' | A ] = P[ B | A'] = 5/11

So the probability in question is find probability : P[ B | A] or... P[ B' | A'], which = 6/11. And that is my final answer. IS this the most efficient way to solve the problem? To me it makes the most sense.