A fair coin is tossed 8 times. Find the probability that it shows heads
(i) exactly 5 times (March ' 17)
(2) at lest once . (March ' 14- 17 )

Explanation

Let X be the number of heads out of 9 tosses .
P (getting head) = ` p = 1/2`
` :. " " q = 1 - p = 1 - 1/2 = 1/2 `
Here, n = 9
` :. " " X ~ B (n,P) = X ~ B (9, 1/2)`
The p.m.f. of X is given by
` P(X = x) = .^(9)C_(x) (1/2)^(x) * (1/2)^(9-x)`
` :.` P (exactly 5 heads) = P (X = 5)
` :. " " P(X = 5) = .^(9)C_(5) * (1/2)^(5) * (1/2)^(9-5)`
` = .^(9)C_(5) (1/2)^(5) (1/2)^(4) `
` = (9!)/(5! 4!) xx 1/2^(9)`
` = (9 x 8 xx 7 xx 6 xx 5 !)/(5 ! xx 4 xx 3 xx 2 xx 1) xx 1/(512)`
` :. " " P(X=5) = 63/(256) `