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} $