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