Question

Let W be a random variable giving the number of

tails

minus the number of

heads

in three tosses of a coin. Assuming that a

tail

is

half

as likely to​ occur, find the probability distribution of the random variable W.

Complete the following probability distribution of W.

Answer 1

When a coin is tossed thrice, S = {hhh, hht, hth, htt, ttt, tth, tht, thh}

Let P(h) = p then P(t) = p/2

p + p/2 = 1, so p = 2/3

Thus, P(h) = 2/3 and P(t) = 1/3

W = number of tails – number of heads

Event	n(t)	n(h)	W = n(t) - n(h)	P(W)
hhh	0	3	-3	(2/3)(2/3)(2/3) = 8/27
hht	1	2	-1	(2/3)(2/3)(1/3) = 4/27
hth	1	2	-1	(2/3)(1/3)(2/3) = 4/27
htt	2	1	1	(2/3)(1/3)(1/3) = 2/27
ttt	3	0	3	(1/3)(1/3)(1/3) = 1/27
tth	2	1	1	(1/3)(1/3)(2/3) = 2/27
tht	2	1	1	(1/3)(2/3)(1/3) = 2/27
thh	1	2	-1	(1/3)(2/3)(2/3) = 4/27

W	P(W)
-3	8/27
-1	4/27 + 4/27 + 4/27 = 12/27
1	2/27 + 2/27 + 2/27 = 6/27
3	1/27