Question

If a fair coin is tossed $5$ times, the probability that heads does not occur two or more times in a row is

• A. $\frac{12}{2^5}$
• B. $\frac{13}{2^5}$
• C. $\frac{14}{2^5}$
• D. $\frac{15}{2^5}$

Answer 1

Answer: (B)

A fair coin is tossed $5$ times.
Total number of sample space $=2^5$
Probability of head does not occur two or more time in rows
Possible cases
Case I All tails $={\left(\frac{1}{2}\right)}^5=\frac{1}{2^5}$
Case II $4$ tails and one head
${}^5{C}_4{\left(\frac{1}{2}\right)}^5=\frac{5}{2^5}$
Case III $3$ tails and $2$ head $\times T\times T\times T\times$
${}^4{C}_2\times {\left(\frac{1}{2}\right)}^5=\frac{6}{2^5}$
Case IV $2$ tails and $3$ heads $\times T\times T\times$
${}^3{C}_3\times {\left(\frac{1}{2}\right)}^5=\frac{1}{2^5}$
$\therefore$ Required probability
$=\frac{1}{2^5}+\frac{5}{2^5}+\frac{6}{2^5}+\frac{1}{2^5}$
$=\frac{13}{2^5}$