Question

Every day you flip a fair coin four times and if it is heads all four times, you give a dollar to charity. In a year with 365 days, what is your expected annual donation to charity and what is the variance?

Answer 1

We have 16 possible outcomes. HHHH, HHHT, HHTH, HTHH, THHH, HHTT, HTHT, HTTH, THTH, TTHH, THHT, TTTH, TTHT, THTT, HTTT and TTTT

Therefore Probability of getting all 4 heads (HHHH) = 1/16 = 0.0625

Probability of getting 3 heads and 1 Tail (HHHT, HHTH, HTHH, THHH) = 4 / 16 = 0.25

Probability of getting 2 heads and 2 Tails (HHTT, HTHT, HTTH, THTH, TTHH, THHT) = 6 / 16 = 0.375

Probability of getting 3 Tails and 1 Head (TTTH, TTHT, THTT, HTTT) = 4 / 16 = 0.25

Probability of getting all 4 Tails (TTTT) = 1 / 16 = 0.0625

Probability distribution is as below

	All Heads	3H and 1T	2H and 2T	3T and 1 H	All Tails
P(x)	0.0625	0.25	0.375	0.25	0.0625
x	1	0	0	0	0

The expected value for 1 day = Sum[x * P(x)] = (1 * 0.0625) + (0*0.25) + (0.*0.375) + (0.*0.25) + (0.*0.0625) = 0.0625

Therefore Annual donated amount = 365 * 0.0625 = 22.8125

Variance = Sum[x² * P(x)] - (E)²

(1² * 0.0625) + (0²*0.25) + (0².*0.375) + (0².*0.25) + (0².*0.0625) - (0.0625)² = 0.0625 - 0.00390625 = 0.05859375

Therefore the variance in a year = 0.05859375 * 365 = 21.3867