Find the Probability Distribution of the Number of Doublets in Three Throws of a Pair of Dice and Find Its Mean. - Mathematics

Question

Find the probability distribution of the number of doublets in three throws of a pair of dice and find its mean.

Solution

Total number of outcomes when two dice are thrown = 6 x 6, i.e. 36
Let X be the number of doublets in three throws of a pair of dice.
Then, X follows a binomial distribution with n = 3

$p = P(\text{ Getting a doublet in three throws } ) = \frac{6}{36} = \frac{1}{6} \text{ and } q = \frac{5}{6}$
$\therefore P(X = r) = ^ {3}{}{C}_r \left( \frac{1}{6} \right)^r \left( \frac{5}{6} \right)^{3 - r} , r = 0, 1, 2, 3$
$\text{ Mean }\left( \text{ np } \right) = 3\left( \frac{1}{6} \right) = \frac{1}{2}$

$\text{ The distribution is as follows } :$
X        0             1                 2                 3
$P(X) \left( \frac{5}{6} \right)^3 \ 3 \left( \frac{1}{6} \right)^1 \left( \frac{5}{6} \right)^{3 - 1} 3 \left( \frac{1}{6} \right)^2 \left( \frac{5}{6} \right)^{3 - 2} \left( \frac{1}{6} \right)^3$
$\frac{125}{216} \ \frac{75}{216} \frac{15}{216} \frac{1}{216}$

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APPEARS IN

RD Sharma Solution for Mathematics for Class 12 (Set of 2 Volume) (2018 (Latest))
Chapter 33: Binomial Distribution
33.2 | Q: 21 | Page no. 25

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Find the Probability Distribution of the Number of Doublets in Three Throws of a Pair of Dice and Find Its Mean. Concept: Random Variables and Its Probability Distributions.