Two numbers are selected at random (without replacement) from the… - Vidyadip

Question

Two numbers are selected at random $($without replacement$)$ from the first six positive integers. Let $X$ denote the larger of the two numbers obtained. Find the probability distribution of the random variable $X,$ and hence find the mean of the distribution.

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Answer

Total number of ways of selecting two numbers$={ }^2 \mathrm{C}_6=15$
Values of $x ($larger of the two$)$ can be $2, 3, 4, 5, 6$
$\text{ P(x = 2) } = \frac{1}{15},\text{ P(x = 3)} = \frac{2}{15},\text{ P(x = 4)} = \frac{3}{15}$
$\text{P(x = 5)} = \frac{4}{15}\text{ and P(x = 6)} = \frac{5}{15}$
$\therefore$ Distribution can bewritten as
$x: 2 3 4 5 6$
$P(x): \frac{1}{15}\frac{2}{15}\frac{3}{15}\frac{4}{15}\frac{5}{15}$
$x \ P(x): \frac{2}{15}\frac{6}{15}\frac{12}{15}\frac{20}{15}\frac{30}{15}$
$\text{ Mean } = \sum\text{x}\text{ P(x)} = \frac{70}{15}= \frac{14}{3}.$

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