Probability question. What is the probability that 2 people out of a group....?

Question

of thirty will have their birthday on the same month and day?

I've heard there is a high probability but I lack the math knowledge to confirm it. If you know the answer could you show how one would calculate it?

Answer 1

There is a high probability. The best way to calculate it is to work out the probability that NONE of your 30 people will have the same birthday.

For the purpose of this calculation, I'll assume that a year has 365 days, and those people born on 29 February, are just... problematic.

Start with person A. You ask him his birthday, and cross it off your calendar. All fine so far - you CAN'T have any clashes yet.

Now person B - the chance that her birthday is different from person A's is 364/365, because there are 364 days left. You cross that one off as well.

So far we've got 1 x 364/365

Now ask person C. His birthday has to be different to either A's or B's. The chance of that is 363/365.

As you can see, the fewer blank spaces there are in your calendar, the more likely it is that the next person you ask is going to have a birthday on a date you've already crossed off.

So we've got 1 x 364/365 x 363/365

and so on, until you've got

1 x 364/365 x 363/365 x 362/365 x 361/365, etc, etc, etc.... x 336/365

This works out to be 0.29368

So the chance that no-one will have the same birthday is 29.37%

Conversely, the chance that there will be at least one "clash" is 70.63%

(My birthday is the same as the person who sits next to me, in an office of roughly thirty people. It's very useful: on the relevant day, we demanded and got ice-cream cake).

Answer 2

The probability that two people out of a group of 30 will have the same birthday is

One less the probability that no two people have the same birthday.

p = 1 - (365*364*363*...336)/365^(30) = 0.7063

It only takes 23 people for the probability to exceed 50%.

This is less people than might seem necessary at first but remember, you haven't specified which two people will match or what the matching birthday will be.

Answer 3

1 out of 15