A bag A contains 5 white and 6 black balls. Another bag B contains 4 white and 3 black balls. A ball is transferred from bag A to the bag B, and then a ball is taken out of the second bag. Find the probability of this ball being black.

Given:

Bag A contains 5 white and 6 black balls.

Bag B contains 4 white and 3 black balls.

A ball is transferred from bag A to bag B, and then a ball is drawn from bag B.

There are two mutually exclusive ways to draw a black ball from bag B –

a. A white ball is transferred from bag A to bag B, and then, a black ball is drawn from bag B

b. A black ball is transferred from bag A to bag B, and then, a black ball is drawn from bag B

Let E₁ be the event that white ball is drawn from bag A and E₂ be the event that black ball is drawn from bag A.

Now, we have

We also have

Let E₃ denote the event that black ball is drawn from bag B.

Hence, we have

We also have

Using the theorem of total probability, we get

P(E₃) = P(E₁)P(E₃|E₁) + P(E₂)P(E₃|E₂)

Thus, the probability of the drawn ball being black is.