A high school question - about the probability of birthday
The day of the week a person is born is random. A large group of people began naming, one at a time, the day of the week that preceded the day they were born, until seven working days were used. When all seven days of the week are first mentioned, what is the expected number of people mentioning each day of the week? Assume that every day is equally possible every step of the way.
Solution: Since the birth of everyone every day is equally possible, then for the first person, the probability that he (she) is born on a certain day is 7/7.
If a person is not born on the birth day of the previous person, the probability is 6/7, and so on, the probability of another person being born on another working day is 5/7, until different seven days of the week are used.
Therefore, the probability of being mentioned for each new day can be extracted as:
where i is the i-th day to be named,
such that the countdown 1/p is the number of times a certain day of the week will appear, that is, it appears The number of people needed on this day.
is the mathematical number of days when the i-th day is mentioned and a new baby is named. The reason why the denominator of
is 7 minus (i-1) is because i-1 days have already been used for naming.
In this way, i=1, 2, 3, 4, 5, 6, 7 are brought in and added together:
This is the expected value of the person who has been used for seven days for the first time.
