Probability 3 Question 12
12. A fair die is rolled. The probability that the first time 1 occurs at the even throw, is
(2005, 1M)
(a) $1 / 6$
(b) $5 / 11$
(c) $6 / 11$
(d) $5 / 36$
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Solution:
- Let $E$ be the event of getting 1 on a die.
$\Rightarrow P(E)=\frac{1}{6}$ and $P(\bar{E})=\frac{5}{6}$
$\therefore \quad P$ (first time 1 occurs at the even throw)
$$ =t _2 \text { or } t _4 \text { or } t _6 \text { or } t _8 \ldots \text { and so on } $$
$={P(\bar{E}) P(E)}+{P(\bar{E}) P(\bar{E}) P(\bar{E}) P(E)}+\ldots \infty$
$=\frac{5}{6} \frac{1}{6}+\frac{5}{6}^{3} \frac{1}{6}+\frac{5}{6}^{5} \frac{1}{6}+\ldots \infty=\frac{\frac{5}{36}}{1-\frac{25}{36}}=\frac{5}{11}$
