Question

Two balls are drawn at random with replacement from a box containing 10 black and 8 red balls. Find the probability that:
A) Both balls are red.
B) First ball is black and the second is red.
C) One of them is black and the other is red.

Answer 1

Hint: An event ‘A’ associated with a random experiment is said to occur if any one of the elementary events A outcome of a random experiment is called an elementary event. Associated to the event is an outcome.
If there are $n$ elementary events associated with a random experiment and $m$ of them are favorable to an event A, then the probability of happening or occurrence of event A is denoted by $P(A)$ and is defined as the ratio $\dfrac{m}{n}$.
Thus, $P(A)$=$\dfrac{m}{n}$.

Complete step-by-step answer:
Total number of balls in a box=18
Number of red balls in a box=8.
Number of black balls in a box=10
Two balls are drawn with replacement from a box.
(i) To find the probability that both the balls are red.
i.e. P(both balls are red)=P(first ball is red)XP(second ball is red given first is red)
  = $\dfrac{8}{{18}} \times \dfrac{8}{{18}}$
After simplifying, we get $\dfrac{4}{9} \times \dfrac{4}{9} = \dfrac{{16}}{{81}}$
(ii) To find the probability that the first ball is black and the second is red.
i.e. P(First ball is black and second is red)=P(first ball is black)XP(second ball is red given first is black)
$ = \dfrac{{10}}{{18}} \times \dfrac{8}{{18}}$
After simplifying, we get
$ = \dfrac{5}{9} \times \dfrac{4}{9}$ \[\dfrac{5}{9} \times \dfrac{4}{9} = \dfrac{{20}}{{81}}\].
(iii) To find the probability that one of them is black and the other is red.
i.e. P(first ball is black & second is red)XP(first ball is red & second is black)
=$\left( {\dfrac{{10}}{{18}} \times \dfrac{8}{{18}}} \right) + \left( {\dfrac{8}{{18}} \times \dfrac{{10}}{{18}}} \right)$
After simplifying, we get $2 \times \left( {\dfrac{8}{{18}} \times \dfrac{{10}}{{18}}} \right) = \dfrac{{40}}{{81}}$.

Note: A random experiment is an experiment in which;
All the outcomes of the experiment are known in advance, and
The exact outcome of any specific performance of the experiment is unpredictable, i.e., are not known in advance.

The set of all possible outcomes of a random experiment is known as sample space associated with the random experiment. An event is something that happens.