Question

If you buy a lottery ticket in 50 lotteries, in each of which your chance of winning a prize is \(\frac{1}{100}\), what is the (approximate) probability that you will win a prize (a) at least once, (b) exactly once, (c) at least twice?

Short answer

The approximate probabilities are (a) \(P_{atleast1} \approx 0.395\), (b) \(P_{exactly1} \approx 0.361\), and (c) \(P_{atleast2} \approx 0.034\).

Step-by-step solution

Find complementary probability of not winning a prize

In each lottery, the probability of winning a prize is 1/100. Therefore, the probability of not winning a prize in a single lottery is the complementary probability: \( P_{notwin} = 1 - P_{win} \) \( P_{notwin} = 1 - \frac{1}{100} \) Compute the numerical value of the complementary probability.

(a) At least once

To find the probability of winning a prize at least once in 50 lotteries, we can use the complementary probability rule. Calculate the probability of not winning any prizes in all 50 lotteries and then subtract from 1. \(P_{atleast1} = 1 - P_{notwin}^{50}\) Compute the approximate probability of winning at least one prize in 50 lotteries.

(b) Exactly once

Use the binomial probability formula to find the probability of winning exactly one prize in 50 lotteries: \(P_{exactly1} = \binom{50}{1} (P_{win})^{1} (P_{notwin})^{49}\) Compute the approximate probability of winning exactly one prize in 50 lotteries.

(c) At least twice

To find the probability of winning at least two prizes in 50 lotteries, we can calculate its complementary probability and subtract it from 1. The complementary probability in this case is winning exactly 0 prizes or 1 prize. \(P_{atleast2} = 1 - (P_{exactly0} + P_{exactly1})\) \(P_{exactly0} = \binom{50}{0} (P_{win})^{0} (P_{notwin})^{50}\) Compute the probability of winning at least two prizes in 50 lotteries.