Q. 4.39 - A First Course in Probability

Question

A ball is drawn from an urn containing $3$ white and $3$ black balls. After the ball is drawn, it is replaced and another ball is drawn. This process goes on indefinitely. What is the probability that the first $4$ balls drawn, exactly $2$ are white?

Step-by-Step Solution

Answer

The probability that of the first $4$ balls drawn, exactly $2$ are white is $\frac{3}{8}.$

Step 1: Given Information

Given in the question that a ball is drawn from an urn containing $3$ white and $3$ black balls. After the ball is drawn, it is replaced and another ball is drawn.

Step 2: Solution of the Problem

Probability of drawing white ball$=\frac{3}{6}$

$=\frac{1}{2}$

Probability of drawing a black ball $=\frac{3}{6}$

$=\frac{1}{2}$

Number of ways of choosing $2$ white  $4$ draws $=\left(\begin{array}{l}4\\ 2\end{array}\right)$

$=6 ways$

Desired probability$=6\times {\left(\frac{1}{2}\right)}^2\times {\left(\frac{1}{2}\right)}^2$

$=\frac{3}{8}.$

Step 3: Final Answer

The probability that of the first 4 balls drawn, exactly $2$ are white is $\frac{3}{8}.$