Ex 13.5, 10

Transcript

Ex 13.5, 10 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 (a) at least once (b) exactly once (c) at least twice? Let X : Number of times he wins a prize Winning a prize on lottery is a Bernoulli trial So, X has a binomial distribution P(X = x) = nCx 𝒒﷮𝒏−𝒙﷯ 𝒑﷮𝒙﷯ n = number of lotteries = 50 p = Probability of winning a prize = 1﷮100﷯ q = 1 – p = 1 − 1﷮100﷯ = 99﷮100﷯ Hence, ⇒ P(X = x) = 50Cx 𝟏﷮𝟏𝟎𝟎﷯﷯﷮𝒙﷯ 𝟗𝟗﷮𝟏𝟎𝟎﷯﷯﷮𝟓𝟎−𝒙﷯ (a) Probability that he wins the lottery atleast once P (at least once) = P(X ≥ 1) = 1 − P (0) = 1 − 50C0 1﷮100﷯﷯﷮0﷯ 99﷮100﷯﷯﷮50−0﷯ = 1 − 1 × 1 × 99﷮100﷯﷯﷮50﷯ = 1 − 99﷮100﷯﷯﷮50﷯ (b) Probability that he wins the lottery exactly once P (exactly once) = P(X = 1) = 50C1 1﷮100﷯﷯﷮1﷯ 99﷮100﷯﷯﷮50−1﷯ = 50 × 1﷮100﷯ × 99﷮100﷯﷯﷮49﷯ = 1﷮2﷯ 99﷮100﷯﷯﷮49﷯ (c) Probability that he wins the lottery atleast twice P (atleast twice) = P(X ≥ 2) = 1 – [P(X = 0) + P(X = 1)] = 1 – 50C0 1﷮100﷯﷯﷮0﷯ 99﷮100﷯﷯﷮50−0﷯+ 50C1 1﷮100﷯﷯﷮1﷯ 99﷮100﷯﷯﷮50−1﷯﷯ = 1 – 99﷮100﷯﷯﷮50﷯+ 1﷮2﷯ 99﷮100﷯﷯﷮49﷯﷯ = 1 – 99﷮100﷯﷯﷮49﷯ 99﷮100﷯+ 1﷮2﷯﷯ = 1 – 99﷮100﷯﷯﷮49﷯ 99 + 50﷮100﷯﷯ = 1 − 149﷮100﷯ 99﷮100﷯﷯﷮49﷯