Question

Determine the probability that in a group of 5 people, at least two share the same birth month. Assume that all 12 months are equally likely to be someone’s birth month.

a) How many choices are there for the birth months of these 5 people (without any restrictions)?

b) How many choices are there for the 5 people to have all different birth months?

c) Report the probability that in a group of 5 people, at least two share the same birth month. (Report your final answer as a decimal, as well as showing how you reach that answer.)

d) Is it reasonable to assume that the 12 months of the year are equally likely to be a person’s birth month? Explain briefly.

Answer 1

(a)

Let the 5 people be A, B, C, D and E.

Number of choices for the birth months of these 5 people (without any restrictions) = 12⁵ = 248832

So,

Answer is:

248832

(b)

Person A has got choices = 12 months, since he can be born on any of the 12 months.

Person B has got choices = 11 months, since he can be born on other than A.

Person C has got choices = 10 months, since he can be born on other than A.and B.

Person D has got choices = 9 months, since he can be born on other than A, B and C.

Person E has got choices = 8 months, since he can be born on other than A, B, C and D.

Thus,

Number of choices for the 5 people to have all different birth months = 12 X 11 X 10 X 9 X 8 = 95040

So,

Answer is:

95040

(c)

The probability that in a group of 5 people, at least two share the same birth month is given by:

So,

Answer is:

0.6181

(d)

It is not reasonable to assume that the 12 months of the year are equally likely to be a person’s birth month because the total number of days are not the same for all months: January, March, May, July, August, October and December have 31 days. April, June, September and November have 30 days. February has 28 or 29 depending on non-leap year or leap year.