Question

A fair die is rolled. The probability that the first time 1 occurs at an even throw, is

• A. $\frac{1}{6}$
• B. $\frac{5}{11}$
• C. $\frac{6}{11}$
• D. $\frac{5}{36}$

Answer 1

The correct option is B $\frac{5}{11}$

Let E be the event of getting 1 in a die.
$\Rightarrow P\left(E\right)=\frac{1}{6}andP\left(\overline{E}\right)=\frac{5}{6}$
$\therefore$ P(first time 1 occurs at the even throw)
$=t_{2}or{t}_{4}or{t}_{6}or{t}_8$ . . . and so on
$=\left\{P\right(\overline{E}\left)P\left(E\right)\right\}+\left\{P\right(\overline{E}\left)P\left(\overline{E}\right)P\left(\overline{E}\right)P\right(E\left)\right\}+\cdots \infty =\left(\frac{5}{6}\right)\left(\frac{1}{6}\right)+{\left(\frac{5}{6}\right)}^3\left(\frac{1}{6}\right)+{\left(\frac{5}{6}\right)}^5\left(\frac{1}{6}\right)+\cdots \infty =\frac{\frac{5}{36}}{1-\frac{25}{36}}=\frac{5}{11}$