Probability 1 Question 2
2. 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) $\frac{175}{6^{5}}$
(b) $\frac{225}{6^{5}}$
(c) $\frac{200}{6^{5}}$
(d) $\frac{150}{6^{5}}$
(2019 Main, 12 Jan I)
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Answer:
Correct Answer: 2. (a)
Solution:
- Since, the experiment should be end in the fifth throw of the die, so total number of outcomes are $6^{5}$.
Now, as the last two throws should be result in two fours (i) $\frac{\text { (ii) }}{\text { (iii) }} \frac{4}{(\text { iv) }} \frac{4}{(v)}$
So, the third throw can be 1, 2, 3, 5 or 6 (not 4). Also, throw number (i) and (ii) can not take two fours in succession, therefore number of possibililites for throw (i) and (ii) $=6^{2}-1=35$
$[\because$ when a pair of dice is thrown then $(4,4)$ occur only once].
Hence, the required probability $=\frac{5 \times 35}{6^{5}}=\frac{175}{6^{5}}$
