Use Poisson approximations to investigate the following type

Use Poisson approximations to investigate the following types of coincidences. The usual assumptions of the birthday problem apply, such as that there are 365 days in a year, with all days equally likely. (a) How many people are needed to have a 50% chance that at least one of them has the same birthday as you? (b) How many people are needed to have a 50% chance that there are two people who not only were born on the same day, but also were born at the same hour (e.g., two people born between 2 pm and 3 pm are considered to have been born at the same hour).

Solution

a) Prob atleast one more person has the same birthday

=50% = 0.5

We have to find the no of persons

p for a person to have birth day on a particular day = 1/365

If n is the no of persons, then Poisson parameter = n/365

Poisson prob for x>=2, is 0.5 means parameter = 1.67865

i.e. n/365 = 1.67865

n = 612.70 = 613

b) For same hour p =1/24 for each

Hence for lemda as previous 1.67865 for two people

1.678675 = n/365*24

or n = 14712 people