If you buy a lottery ticket in 50 lotteries, in each of...

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?

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Key Concepts

Summing Binomial Probabilities

Summing binomial probabilities involves adding the probabilities of individual outcomes to determine the probability of a range of successes. This method is used to calculate probabilities for events such as 'exactly once' or 'at least twice' by summing the appropriate terms provided by the binomial distribution or by subtracting the sum of the undesired outcomes from 1.

Binomial Distribution

The binomial distribution is a fundamental concept in probability that models the number of successes in a fixed number of independent trials, where each trial has the same probability of success. It provides the framework to compute probabilities for exactly k successes using the formula: P(X = k) = (n choose k) * p^k * (1-p)^(n-k), which is directly applicable to dice, lottery draws, and similar situations.

Complement Rule

The complement rule is a key principle in probability that states the probability of an event occurring is 1 minus the probability of it not occurring. This concept is especially useful in scenarios such as calculating the probability of winning at least once, where it is often simpler to first compute the probability of not winning at all and then subtract that result from 1.

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