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(A) The repeated tosses of a pair of dice are Bernoulli trials. Let $X$ denote the number of times of getting doublets in an experiment of throwing tw

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$\frac{25}{216}$

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(A) The repeated tosses of a pair of dice are Bernoulli trials. Let $X$ denote the number of times of getting doublets in an experiment of throwing two dice simultaneously $4$ times. Probability of getting doublets in a single throw of the pair of dice is $p = \frac{6}{36} = \frac{1}{6}$. Therefore,$q = 1 - p = 1 - \frac{1}{6} = \frac{5}{6}$. Clearly,$X$ has the binomial distribution with $n = 4$,$p = \frac{1}{6}$,and $q = \frac{5}{6}$. The probability of $x$ successes is given by $P(X = x) = ^{n}C_{x} q^{n-x} p^{x}$. For $x = 2$: $P(X = 2) = ^{4}C_{2} \cdot (\frac{5}{6})^{4-2} \cdot (\frac{1}{6})^{2}$ $P(X = 2) = 6 \cdot (\frac{5}{6})^{2} \cdot (\frac{1}{6})^{2}$ $P(X = 2) = 6 \cdot \frac{25}{36} \cdot \frac{1}{36}$ $P(X = 2) = 6 \cdot \frac{25}{1296} = \frac{25}{216}$.

$A$ pair of dice is thrown $4$ times. If getting a doublet is considered a success,find the probability of two successes.

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