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. 

Answer 1

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`