Question

Two numbers are selected at random (without replacement) from the first six positive integers. Let X denote the larger of the two numbers. Find E(X).

Answer 1

Two number are selected without replacement from {1, 2, 3, 4, 5, 6}.

Let S = sample space

∴ n(S) = ⁶C₂

= `(6!)/(2! xx 4!)`

= `(6 xx 5 xx 4!)/(2 xx 1 xx 4!)`

= 15

Let X denote the larger of the two numbers obtained.

∴ Possible values of X are 2, 3, 4, 5, 6.

When X = 2,
one of the two numbers is 2 and remaining one is smaller than 2, i.e., 1.

∴ Remaining number can be selected in 1 way only.

∴ n(X = 2) = 1

∴ P(X = 2) = `(1)/(15)`

When X = 3,

one of the two numbers is 3 and remaining one is smaller than 3, i.e., 1 or 2.

∴ Remaining number can be selected in ²C₁ = 2 ways.

∴ n(X = 3) = 2

∴ P(X = 3) = `(2)/(15)`

Similarly, P(X = 4) = `(3)/(15)`, P(X = 5) = `(4)/(15)`, P(X = 6) = `(5)/(15)`

∴ E(X) = \[\sum\limits_{i=1}^{5} x_i.\text{P}(x_i)\]

= `2 xx (1)/(15) + 3 xx (2)/(15) + 4 xx (3)/(15) + 5 xx (4)/(15) + 6 xx (5)/(15)`

= `(1)/(15)(2 + 6 + 12 + 20 + 30)`

= `(14)/(3)`

= 4.67