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Probability — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsProbability5 Marks
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= $^2C_6 = 15$
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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