Question

A fair coin is tossed repeatedly until two consecutive heads are obtained. If the probability that $2$ consecutive heads occur on fourth and fifth toss is $p$ , then $\frac{30}{p}$ is equal to

Answer 1

Fourth and fifth toss turns up head hence tail must occur on third toss.
So, $HT,TH$ or $TT$ occurs on the first and second toss.
Hence, the required probability $=\underset{\left(H T T H H\right)}{\underbrace{\frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2}}}+\underset{\left(T H T H H\right)}{\underbrace{\frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2}}}+\underset{\left(T T T H H\right)}{\underbrace{\frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2}}}$
$=\frac{3}{32}$