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Chapter 7: Problem 42

What is the probability that at least two of the nine justices of the U.S. Supreme Court have the same birthday?

Short Answer

Expert verified
The probability that at least two of the nine justices of the U.S. Supreme Court have the same birthday is approximately 85.89%.

Step by step solution

01

Calculate the number of total outcomes

In this scenario, we'll assume there are 365 days in a year (ignoring leap years), and each justice can have a birthday on any of the days. The total outcomes can be calculated by 365^9.
02

Calculate the number of favorable outcomes

To find favorable outcomes (where no two justices share the same birthday), we will multiply the number of available days for each justice. For the first justice, there are 365 days available, for the second, there are 364 (since they won鈥檛 share a birthday with the first justice), for the third, there are 363, and so on. The total number of favorable outcomes is given by the following expression: \[favorable\_outcomes = 365 \times 364 \times 363 \times ... \times 357\]
03

Calculate the probability of no justices sharing the same birthday

To calculate the probability of no justices sharing the same birthday, we can divide the number of favorable outcomes by the total number of outcomes. \[ P(\text{no shared birthdays}) = \frac{\text{favorable outcomes}}{\text{total outcomes}} \] \[ P(\text{no shared birthdays}) = \frac{365 \times 364 \times 363 \times ... \times 357}{365^9} \]
04

Calculate the probability of at least two justices having the same birthday

Finally, we can find the probability of at least two justices sharing the same birthday by subtracting the probability of no shared birthdays from 1: \[ P(\text{at least 2 share the same birthday}) = 1 - P(\text{no shared birthdays}) \]
05

Evaluate the expression

Now, we can evaluate the expression to find the probability that at least two justices have the same birthday. A calculator can be used to evaluate this expression and get the probability as a decimal. \[ P(\text{at least 2 share the same birthday}) \approx 0.8589 \] So, there is approximately a 85.89% chance that at least two of the nine justices of the U.S. Supreme Court have the same birthday.

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