# A bag contains 3 white and 2 black balls, and another bag contains 2 white and 4 black balls. One bag is chosen at random. From the selected bag, one ball is drawn. Find the probability that the ball drawn is white.

Given:

The bag I contains 3 white and 2 black balls.

Bag II contains 2 white and 4 black balls.

A bag is chosen, and a ball is drawn from it.

There are two mutually exclusive ways to draw a white ball from one of the two bags –

a. The bag I is selected, and then, a white ball is drawn from the bag I

b. Bag II is selected, and then, a white ball is drawn from bag II

Let E1 be the event that bag I is selected and E2 be the event that bag II is selected.

Since there are only two bags and each bag has an equal probability of being selected, we have

Let E3 denote the event that a white ball is drawn.

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 white is.

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