Assume there are 40 students in a class each born on one of

Assume there are 40 students in a class, each born on one of the 365 days in a non-leap year. Write expressions for the following probabilities:

a) The probability that at least one student in the class (other than you) has the same birthday as you.

b) The probability that there is a pair of distinct students in the class with the same birthday.

Solution

a)

P(at least one) = 1 - P(none has same bairthday)

My 39 classmates can choose their birthdays in 365^39 ways. Thus, if none has the same birthday as mine, they can choose in 364^39 ways.

thus,

P(at least one) = 1 - (364^39)/(365^39) = 0.10147069 [ANSWER]

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b)

P(there is a pair with same bday) = 1 - P(all have distinct bdays)

There are 365P40 ways so that all have distinct birthdays. Thus,

P(there is a pair with same bday) = 1 - (365P40)/(365^40) = 0.89123181 [ANSWER]