Question

A fair coin is tossed 3 times. A person receives ₹ X² if he gets X number of heads in all. Find his expected gain.

Answer 1

Here S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}

∴ n(S) = 8

Let X = Number of heads.

∴ X = {0, 1, 2, 3}

∴ P(X = 0) = `1/8`,

P(X = 1) = `3/8`,

P(X = 2) = `3/8`,

P(X = 3) = `1/8`

The probability distribution of X is

X	0	1	2	3
P(X)	`1/8`	`3/8`	`3/8`	`1/8`

Now for X heads the man received Rs. X²

∴ The probability distribution for the amount received is

X (₹)	0	1	4	0
P(X = x)	`1/8`	`3/8`	`3/8`	`1/8`

E(X) = `sumx_i*p_i`

= `(0)(1/8) + (1)(3/8) + (4)(3/8) + (9)(1/8)`

= `0 + 3/8 + 12/8 + 9/8`

= `24/8`

= 3

∴ Expected gain = ₹ 3