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Chapter 7: Problem 43

Fifty people are selected at random. What is the probability that none of the people in this group have the same birthday?

Short Answer

Expert verified
The probability that none of the 50 people have the same birthday is approximately 0.03 or 3%. This is calculated using the formula \( \frac{365! / (365-50)!}{365^{50}}\).

Step by step solution

01

Determine Possible Outcomes

There are 365 days in a year, thus a person can have their birthday on any of these days. As we don't have any restrictions on multiple people having the same birthday, for each of the 50 people there are 365 possibilities. Hence, the total number of ways for 50 people to have birthdays is \(365^{50}\).
02

Determine Desired Outcomes

In figuring the number of arrangements where all 50 people have different birthdays, for the first person there are 365 choices, for the next person there would be 364 choices left (as their birth date must be different from the first person's), for the third person there would be 363 choices, and so on until the 50th person for whom there'd be 316 choices remaining. This results in total arrangements with different birthdays as \(365*364*363*...*316\).
03

Calculate the Probability

The probability of an event is the number of ways the event can happen divided by the total number of possible outcomes. Hence, the probability that no two people out of the 50 have the same birthday is \(\frac{365*364*363*...*316}{365^{50}}\). It's practical to use factorial notation and write this result as \( \frac{365! / (365-50)!}{365^{50}}\).
04

Simplify the Result

To get the final answer, we'll simplify the result calculated in the previous step. This involves a bit of calculation which is easier done with a calculator. The approximation of this probability is about 0.03, or 3% probability that none of the 50 people have the same birthday.

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