Forty people are selected at random. What is the probability that none of the people in this group have the same birthday?
This is a famous problem
https://en.wikipedia.org/wiki/Birthday_problem
To calculate the probability that none of the people in a group have the same birthday, we need to consider the number of possible outcomes where none of the birthdays are the same, and divide it by the total number of possible outcomes.
Let's break down the problem step by step:
Step 1: Determine the total number of possible outcomes.
In this case, we are considering the birthdays of 40 people. Each person has 365 possible birthdays, assuming that we are not considering leap years. Thus, the total number of possible outcomes is 365^40.
Step 2: Determine the number of favorable outcomes.
For none of the people in the group to have the same birthday, each person must have a unique birthday. In other words, the first person can have any of the 365 possible birthdays, the second person can have any of the remaining 364 possible birthdays, the third person can have any of the remaining 363 possible birthdays, and so on. The number of favorable outcomes can be calculated as 365 * 364 * 363 * ... * (365 - (40 - 1)).
Step 3: Calculate the probability.
To find the probability, we divide the number of favorable outcomes by the total number of possible outcomes:
Probability = Number of favorable outcomes / Total number of possible outcomes
Probability = (365 * 364 * 363 * ... * (365 - (40 - 1))) / (365^40)
By calculating this expression, we can find the probability that none of the people in the group have the same birthday.