Question

A fair coin is tossed 8 times. Find the probability that it shows heads exactly 5 times.

Answer 1

Let X = Number of heads

p = probability of getting head in one toss

p = `1/2`

q = `1 - p = 1 - 1/2 = 1/2`

Given n = 8

`x ~ B(8, 1/2)`

The p.m.f. of X is given as

P(X = x) = `""^nC_xp^xq^(n - x)`

i.e P(x) = `""^8C_x(1/2)^x(1/2)^(8 - x), x = 0, 1, 2, 3,......,8`

P(exactly 5 heads) = P[X = 5]

= P(5)

= `""^8C_5(1/2)^5(1/2)^(8 - 5)`

= `""^8C_3(1/2)^5(1/2)^3   ...[because ""^nC_x = ""^nC_(n - x)]`

= `(8 * 7 * 6)/6 xx (1/2)^8`

= `8 xx 7 xx 1/(16 xx 16)`

= `7/32`

∴ P[X = 5] = 0.21875

Hence, the probability of getting exactly 5 heads is 0.21875