A pair of dice is thrown 5 times. For each throw, a total of 5 is considered a success. If the probability of at least 4 successes is , then is equal to [2023]

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Solution

Q.
A pair of dice is thrown 5 times. For each throw, a total of 5 is considered a success. If the probability of at least 4 successes is $\frac{k}{3^{11}}$ , then $k$ is equal to [2023]

1 82

2 164

3 75

4 123

Ans.
(4)

For a total of 5 we have $(1, 4), (4, 1), (2, 3), (3, 2)$

$So, probability of success = \frac{4}{36} = \frac{1}{9} = p$

$and probability of failure = \frac{8}{9} = q$

Now, $P (X \geq 4) = {}^{5}{C_{4}} {(\frac{1}{9})}^{4} \cdot {(\frac{8}{9})}^{1} + {}^{5}{C_{5}} {(\frac{1}{9})}^{5}$

$\Rightarrow \frac{k}{3^{11}} = \frac{40}{9^{5}} + \frac{1}{9^{5}} \Rightarrow k = 41 \times 3 = 123$

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