A pair of fair dice is thrown 4 times. If get | vedclass

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(A) When a pair of dice is thrown,the total number of outcomes is $6 	imes 6 = 36$. Success is defined as getting the same number on both dice. The f

Vedclass · Accepted Answer

$\frac{25}{216}$

Vedclass · Answer

(A) When a pair of dice is thrown,the total number of outcomes is $6 	imes 6 = 36$. Success is defined as getting the same number on both dice. The favorable outcomes are $(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)$,which are $6$ outcomes. Thus,the probability of success $p = \frac{6}{36} = \frac{1}{6}$. The probability of failure $q = 1 - p = 1 - \frac{1}{6} = \frac{5}{6}$. We use the binomial distribution formula $P(X = k) = \binom{n}{k} p^k q^{n-k}$,where $n = 4$ and $k = 2$. $P(X = 2) = \binom{4}{2} \left(\frac{1}{6}ight)^2 \left(\frac{5}{6}ight)^{4-2}$. $P(X = 2) = 6 	imes \left(\frac{1}{36}ight) 	imes \left(\frac{25}{36}ight)$. $P(X = 2) = 6 	imes \frac{25}{1296} = \frac{25}{216}$.

$A$ pair of fair dice is thrown $4$ times. If getting the same number on both dice is considered as a success,then the probability of exactly two successes is:

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