If one ball is drawn at random from each of the three boxes containing $$3$$ white and $$1$$ black, $$2$$ white and $$2$$ black, $$1$$ white and $$3$$ black balls then the probability that $$2$$ white and $$1$$ black balls will be drawn is :
A.
$$\frac{{13}}{{32}}$$
B.
$$\frac{1}{4}$$
C.
$$\frac{1}{{32}}$$
D.
$$\frac{1}{{16}}$$
Answer :
$$\frac{{13}}{{32}}$$
Solution :
Let $${E_1} = $$ the event of drawing a white ball from the first box.
Similarly, $${E_2}$$ and $${E_3}$$
Here $$P\left( {{E_1}} \right) = \frac{3}{4},\,\,P\left( {{E_2}} \right) = \frac{1}{2},\,\,P\left( {{E_3}} \right) = \frac{1}{4}$$
The required probability
$$\eqalign{
& = P\left( {{E_1}{E_2}\overline {{E_3}} } \right) + P\left( {{E_1}\overline {{E_2}} {E_3}} \right) + P\left( {\overline {{E_1}} {E_2}{E_3}} \right) \cr
& = P\left( {{E_1}} \right).P\left( {{E_2}} \right).P\left( {\overline {{E_3}} } \right) + P\left( {{E_1}} \right).P\left( {\overline {{E_2}} } \right).P\left( {{E_3}} \right) + P\left( {\overline {{E_1}} } \right).P\left( {{E_2}} \right).P\left( {{E_3}} \right) \cr
& = \frac{3}{4}.\frac{1}{2}.\frac{3}{4} + \frac{3}{4}.\frac{1}{2}.\frac{1}{4} + \frac{1}{4}.\frac{1}{2}.\frac{1}{4} = \frac{{13}}{{32}}. \cr} $$
Releted MCQ Question on Statistics and Probability >> Probability
Releted Question 1
Two fair dice are tossed. Let $$x$$ be the event that the first die shows an even number and $$y$$ be the event that the second die shows an odd number. The two events $$x$$ and $$y$$ are:
Two events $$A$$ and $$B$$ have probabilities 0.25 and 0.50 respectively. The probability that both $$A$$ and $$B$$ occur simultaneously is 0.14. Then the probability that neither $$A$$ nor $$B$$ occurs is
The probability that an event $$A$$ happens in one trial of an experiment is 0.4. Three independent trials of the experiment are performed. The probability that the event $$A$$ happens at least once is
If $$A$$ and $$B$$ are two events such that $$P(A) > 0,$$ and $$P\left( B \right) \ne 1,$$ then $$P\left( {\frac{{\overline A }}{{\overline B }}} \right)$$ is equal to
(Here $$\overline A$$ and $$\overline B$$ are complements of $$A$$ and $$B$$ respectively).
A.
$$1 - P\left( {\frac{A}{B}} \right)$$
B.
$$1 - P\left( {\frac{{\overline A }}{B}} \right)$$
C.
$$\frac{{1 - P\left( {A \cup B} \right)}}{{P\left( {\overline B } \right)}}$$
D.
$$\frac{{P\left( {\overline A } \right)}}{{P\left( {\overline B } \right)}}$$