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.