Question

There are 5 cards numbered 1 to 5, one number on one card. Two cards are drawn at random without replacement. Let X denote the sum of the numbers on two cards drawn. Find the mean and variance of X.

Answer 1

Here, sample space S = {(1, 2), (2, 1), (1, 3), (3, 1), (2, 3), (3, 2), (1, 4), (4, 1), (1, 5), (5, 1), (2, 4), (4, 2), (2, 5), (5, 2), (3, 4), (4, 3), (3, 5), (5, 3), (5, 4), (4, 5)}

∴ n(S) = 20

Let X be the random variable denoting the sum of the numbers on two cards drawn.

∴ X = 3, 4, 5, 6, 7, 8, 9

So, P(X = 3) = `2/20`

P(X = 4) = `2/20`

P(X = 5) = `4/20`

P(X = 6) = `4/20`

P(X = 7) = `4/20`

P(X = 8) = `2/20`

P(X = 9) = `2/20`

∴  Mean, E(X) = `3 xx 2/20 + 4 xx 2/20 + 5 xx 4/20 + 6 xx 4/20 + 7 xx 4/20 + 8/20 + 9 xx 2/20`

= `6/20 + 8/20 + 20/20 + 24/20 + 28/20 + 16/20 + 18/20`

= `120/20`

= 6

E(X²)  `9 xx 2/20 + 16 xx 2/20 + 25 xx 4/20 + 36 xx 4/20 + 64 xx 2/20 + 81 xx 2/20`

= `18/20 + 32/20 + 100/20 + 144/20 + 196/20 + 128/20 + 162/20`

= `780/20`

= 39

∴ Variance (X)= E(X²) – [E(X)]²

= 39 – (6)²

= 39 – 36

= 3