Question

A box contains ‘a’ white and ‘b’ black balls. ‘c’ balls are drawn at random with replacement. The expected number of white balls drawn is :
(a) $\dfrac{a}{a+b}$
(b) $\dfrac{ac}{a+b}$
(c) $\dfrac{bc}{a+b}$
(d) $\dfrac{b}{a+b}$

Answer 1

Hint: Add the number of white balls and black balls to get the total number of balls. Find the probability of drawing white balls by taking the ratio of the number of white balls with the total number of balls. Use the formula: Expected number of white balls drawn = Probability of white balls drawn $\times $ number of balls drawn at random, to get the answer.

Complete step-by-step solution:
Here, we have been provided with the box containing ‘a’ white balls and ‘b’ black balls.
$\Rightarrow $ Total number of balls in the box = number of (white balls + black balls) = a + b
$\Rightarrow $Probability of white balls drawn = $\dfrac{\text{Number of white balls}}{\text{Total number of balls}}$
Let us assume this probability as $P\left( E \right)$
$\Rightarrow P\left( E \right)=\dfrac{a}{a+b}\ldots \ldots \ldots \left( i \right)$
Now it is given that ‘c’ balls are drawn at random with replacement. We have to find the expected number of white balls drawn. Here, the term replacement means, if we draw a ball, then we do not place the ball separately or remove it but we place it again in the box. Therefore, applying the formula for the expected number, we get,
Expected number of white balls drawn
= Probability of white balls drawn $\times $ number of balls drawn at random
= $P\left( E \right)\times c$
Substituting the value of $P\left( E \right)$ from equation (i), we get,
Expected number $=\dfrac{a}{a+b}\times c=\dfrac{ac}{a+b}$
Hence, option (b) is the correct answer.

Note: One may note that we can also solve this question by assuming and assigning different numerical values to a, b, and c. What we will do is, we will calculate the expected number using these numerical values assigned to a, b, and c, and then we will substitute these values of a, b, and c in the options and match our answer with each option to get the correct one. But remember that this short method is applicable only when options are given. Finally, note that we do not have to use the concept of permutations and combinations here.