Question

A box contains $20$ identical balls of which $10$ are blue and $10$ are green. The balls are drawn at random from the box one at a time with replacement. The probability that a blue ball is drawn $4^{\text{th}}$ time on the $7^{\text{th}}$ draw is

• A. $\frac{27}{32}$
• B. $\frac{5}{64}$
• C. $\frac{5}{32}$
• D. $\frac{1}{2}$

Answer 1

Answer: (C)

Probability of getting a blue ball at any draw
$=p=\frac{10}{20}=\frac{1}{2}$
$P$ [getting a blue ball $4^{\text{th}}$ time in $7^{\text{th}}$ draw]
$=P[$ getting $3$ blue balls in $6$ draw $]\times P$ [a blue ball in the $7^{\text{th}}$ draw]
$={}^6{C}_3{\left(\frac{1}{2}\right)}^3{\left(\frac{1}{2}\right)}^3\cdot \frac{1}{2}$
$=\frac{6\times 5\times 4}{1\times 2\times 3}{\left(\frac{1}{2}\right)}^7$
$=20\times \frac{1}{32\times 4}=\frac{5}{32}$