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प्रश्न
If a fair coin is tossed 10 times and the probability that it shows heads 5 times.
उत्तर
Let, X = number of heads.
p = probability that coin tossed shows a head
∴ p = `1/2`
q = 1 - q = `1 - 1/2 = 1/2`
Given: n = 10
∴ X ~ B `(10, 1/2)`
The p.m.f. of X is given by
P(X = x) = `"^nC_x p^x q^(n - x)`
∴ p(x) = `""^10C_x (1/2)^x (1/2)^(10 - x) = "^10C_x (1/2)^10,` x = 0, 1, 2,...,10
P(coin shows heads 5 times) = P[X = 5]
= p(5) = `"^10C_5 * (1/2)^10`
`= (10!)/(5! * 5!) xx 1/1024`
`= (10 xx 9 xx 8 xx 7 xx 6 xx 5!)/(5! xx 5 xx 4 xx 3 xx 2 xx 1) xx 1/1024 = 63/256`
Hence, the probability that can shows heads exactly 5 times = `63/256`.
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