Question

There are 2 boxes. The first box contains 2 white and 5 black balls. The second box contains 4 white and 7 black balls. A random box is chosen and from it a random ball is drawn. a) What is the probability that the ball is white? b) If the ball is known to be white, what is the probability that it came from the 1st box?

Answer 1

There are 2 boxes.

The first box contains 2 white and 5 black balls.

The second box contains 4 white and 7 black balls.

A random box is chosen; from it a random ball is drawn.

Question (a)

We have to find the probability that the ball is white.

Let,

B1=Event that Box 1 is drawn.

B2=Event that Box 2 is drawn.

W=Event that white ball is drawn.

B=Event that Black ball is drawn.

It is given that

Now, we have to find

By total probability theorem, this becomes

So, the probability that the ball is white is 20/77.

Question (b)

We have to find the probability that the ball came from the first box, given that the ball is white.

So, basically we have to find

By using Bayes' theorem, it becomes

So, the probability that the first ball came from box 1, given that it is known to be white, is 11/20.