Question

In a random experiment, a fair die is rolled until two fours are obtained in succession. The probability that the experiment will end in the fifth throw of the die is equal to:

• A. $\frac{200}{6^5}$
• B. $\frac{150}{6^5}$
• C. $\frac{225}{6^5}$
• D. $\frac{175}{6^5}$

Answer 1

The correct option is D $\frac{175}{6^5}$

The probability that $4$ occurs on the die in a throw is $\frac{1}{6}$ and the probability that any other number occurs is $\frac{5}{6}$.
Let $\overline{4}$ is any number except $4$
The experiment ends at the $5th$ throw therefore, possibility of experiment is
$\overline{4}\overline{4}\overline{4}44,\overline{4}4\overline{4}44,4\overline{4}\overline{4}44$
$\therefore$ Probability $=\frac{5}{6}\times \frac{5}{6}\times \frac{5}{6}\times \frac{1}{6}\times \frac{1}{6}+\frac{5}{6}\times \frac{1}{6}\times \frac{5}{6}\times \frac{1}{6}\times \frac{1}{6}+\frac{1}{6}\times \frac{5}{6}\times \frac{5}{6}\times \frac{1}{6}\times \frac{1}{6}$
$=\frac{175}{6^5}$