For a group of 7 people, find the probability that all 4 seasons (winter, spring, summer, fall) occur at least once each among their birthdays, assuming that all seasons are equally likely

Let be the event that there are no birthdays in the season. The probability that all seasons occur at least once is

Note that

Using the inclusion-exclusion principle and the symmetry of the seasons,

Now find:

Thus,

The probability that all 4 seasons occur at least once is

0.51.

-22

times both sides by 2, times () by 8, add alile terms, move terms, add alike and caculate divide both sides by 3