three men meet by chance What are the probabilities that a n

three men meet by chance. What are the probabilities that (a) none of them, (b) two of them, (c) all of them, have the same birthday?

Solution

Ignoring leap years, there are

365^3 = 48627125 ways

for them to have birthdays.

a)

If none of them have the same birthday, there are 365P3 = 48228180 ways to do that.

Hence,

P(all unique) = 48228180/48627125 = 0.991795834 [answer]

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b)

There are 365 dates that 2 of them can choose as their common birtday.

There are 3C2 = 3 ways to choose which two have the same bday.

There are 364 other ways the third one has a different birthday.

hence, there are 365*3*364 = 398580 ways.

Thus,

P(2 have the same bday) = 398580/48627125 = 0.00819666 [answer]

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c)

There are 365 ways for them to have the same birthdays.

Thus,

P(all the same) = 365/48627125 = 0.0000075061 [answer]