Question

Two numbers are selected at random (without replacement) from the first six positive integers. If $X$ denotes the smaller of the two numbers, then the expectation of $X$, is:

• A. $\frac{14}{3}$
• B. $\frac{13}{3}$
• C. $\frac{7}{3}$
• D. $\frac{5}{3}$

Answer 1

Answer: (C)

$\frac{7}{3}$

Two numbers are selected at random without replacement.

Thus possible pairs are:

$(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)$

Hence, expectation of $X$ will be

$\frac{(1\times 5)+(2\times 4)+(3\times 3)+(4\times 2)+(5\times 1)}{15}=\frac{5+8+9+8+5}{15}=\frac{35}{15}=\frac{7}{3}$