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Question
A person buys a lottery ticket in 50 lotteries, in each of which his chance of winning a prize is `1/100`. What is the probability that he will win a prize at least once.
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Solution
Let X = number of winning prizes.
p = probability of winning a prize
∴ p = `1/100`
and q = 1 − p = 1 − `1/100` = `99/100`
Given: n = 50
∴ X ~ B `(50, 1/100)`
The p.m.f. of X is given by P(X = x) = `""^nC_x p^x q^(n - x)`
i.e. p(x) = `""^50C_x (1/100)^x(99/100)^(50-x), x = 0, 1, 2, ...50`
P(a person wins a prize at least once)
= P[X ≥ 1] = 1 − P[X < 1] = 1 − p(0)
= 1 − `""^50C_0 (1/100)^0 (99/100)^(50-0)`
= 1 − 1 × 1 × `(99/100)^50`
= 1 − `(99/100)^50`
Hence, probability of winning a prize at least once
= 1 − `(99/100)^50`
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