Question

A bag contains 5 white balls and 7 black balls; if two balls are drawn what is the chance that one is white and the other black?

Answer 1

Hint: As the question involves compound events, so firstly we will use compound probability using the formula- P (X and Y) = P(X)$ \times $ P(Y) to find the probability of both the possible cases. Then using the formula of probability of mutually exclusive events i.e. P (X or Y) = P(X) + P(Y) to find the required probability.

Complete step-by-step answer:

As we know that when the events are dependent, then we will find compound probability by given formula -
P (X and Y) = P(X)$ \times $ P(Y)
Where X and Y are two independent events.
In this question, X is white and Y is black.
Here we will consider that the occurrence may be the happening of one or other of the two following compound events:
Drawing a white and then a black ball-
The chance P(X) $ = \dfrac{5}{{12}} \times \dfrac{7}{{11}} = \dfrac{{35}}{{132}}$
Drawing a black and then a white ball-
The chance P(Y) $ = \dfrac{7}{{12}} \times \dfrac{5}{{11}} = \dfrac{{35}}{{132}}$
And when the one or more part of the compound events holds true, then the compound probability is given by-
P (X or Y) = P(X) + P(Y)
The required chance P(X or Y) $ = \dfrac{{35}}{{132}} + \dfrac{{35}}{{132}} = \dfrac{{35}}{{66}}$
Therefore the chance of one is white and the other black = $\dfrac{{35}}{{66}}$.

Note: Here in this question, many of us get confused with the appropriate concept to be used. So we must understand that in this particular question we are not given that whether first is black, the second black or first is black, the second white. Therefore, we will add both chances by using the concept of mutually exclusive events to get required probability.