Question

An unbiased die is rolled until a number greater than $4$ appears. The probability that an even number of trials are needed, is

• A. $\frac{1}{2}$
• B. $\frac{2}{5}$
• C. $\frac{1}{5}$
• D. $\frac{2}{3}$

Answer 1

Answer: (B)

Probability of success $P\left(s\right)=p=\frac{2}{6}=\frac{1}{3}$
Probability of failure $P\left(f\right)=q=1-\frac{1}{3}=\frac{2}{3}$
The probability that success occurs in even number of trials
$=P\left(fs\right)+P\left(fffs\right)+P\left(fffffs\right)+.....$
$=qp+q^{3}p+q^{5}p+\dots =\frac{qp}{1-q^2}$
$=\frac{\frac{2}{3}\times \frac{1}{3}}{1-{\left(\frac{2}{3}\right)}^2}=\frac{\frac{2}{9}}{1-\frac{4}{9}}=\frac{2}{9}\times \frac{9}{5}=\frac{2}{5}$