Use Poisson approximations to investigate the following types of coincidences. T

Question

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).

Explanation / Answer

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

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