Fourty people are selected at random. What is the probability that none of the people in this group have the same birthday?
there are [(40 * 39) / 2] separate pairs of people
the probability of two people not having the same birthday is (364 / 365)
the probability that none of the group have the same birthday is
... (364 / 365)^(20 * 39)
To calculate the probability that none of the people in a group of forty have the same birthday, we need to consider the total number of possible outcomes and the number of favorable outcomes.
There are 365 days in a year (excluding leap years) on which a person can be born. Assuming that birthdays are equally likely to occur on any day, we can calculate the probability as follows:
The first person can have their birthday on any day, so there are 365 possible outcomes for them.
The second person must have their birthday on a different day, so there are 364 possible outcomes for them.
Similarly, the third person must have their birthday on a different day from the first two, so there are 363 possible outcomes for them.
Continuing in this manner, the nth person must have their birthday on a different day from the previous (n-1) people, resulting in (365 - (n-1)) possible outcomes.
So, for a group of forty people, the probability that none of them share the same birthday is:
P(none share same birthday) = (365/365) * (364/365) * (363/365) * ... * (365 - (n-1))/365
Substituting n = 40 into the formula:
P(none share same birthday) = (365/365) * (364/365) * (363/365) * ... * (327/365)
Calculating this product gives us the probability that none of the forty people have the same birthday.