Probability Theory: Coincidental Birthday

Date: 06/21/98 at 22:45:38
From: Ong Ching Hui Rosalind
Subject: Probability theory

What is probability all about? Can this question be solved with other 
methods other than using probability theory?

Coincidental birthday: 

Dr. Brainy, the headmaster of a large comprehensive school, notices 
that in more than half of the classes there are at least two children 
whose birthdays coincide. He argues to himself that as there are 365 
days in a year, there would have to be 366 children with the same 
birthday. He knows that the average class size in his school is 30, so 
he reckons that all those coinciding birthdays must be a record. For 
publicity, he decides to write to the newspaper and the Guinness Book 
of Records. Luckily for him, his colleague Angie Analysis hears of his 
ambitions and stops him, showing that the coincidence is expected. 
What argument does she use? What is the probability of a class of 30 
having at least two people with the same birthday? 

Thank you.

Date: 06/24/98 at 08:54:17
From: Doctor Floor
Subject: Re: Probability theory

Hi Ong Ching Hui Rosalind,

Thank you for sending your question to Dr. Math. It's a nice question.

For the problem of Dr. Brainy and Angie Analysis you'll need 
probability theory. But probability theory is fun!

Let's start with smaller numbers: two people and their birthdays. What 
is the probability that two people, say Jack and John, will have the 
same birthday? We'll check them one by one, to compute the probability 
that the birthday of each one fits into the picture of the birthdays 
we already know.

First look at Jack: there is no birthday to compare with, so the 
probability that Jack fits is 1 (= 100%). It doesn't matter what day 
his birthday is, so all days are "correct."  This might seem an 
unimportant step, but it isn't - many people think that for Jack the 
probability is 1/365, because it has to be John's birthday, but that 
is not correct because you should look *only* at birthdays you have 
already taken into account, and that is not yet one.

Now John: John must have the same birthday as Jack. So there is only 1 
day out of of 365 that makes him fit. That means that the probability 
for Jack is 1/365.

For the total probability that John AND Jack fit, we have to multiply 
the two probabilities (something to remember: AND here means 
MULTIPLY): 1 x 1/365 = 1/365.  So the probability that both John and 
Jack have the same birthday is 1/365. Very small.

Let's build up slowly now to three persons: Joanne, Jackie and Joelle. 
Again we will look one by one to see if at least two of them should 
have the same birthday. Now we have a problem: Joanne fits, but then 
Jackie might or might not have the same birthday, so she fits too. 
But then we don't know what the probability for Joelle is, because it 
depends on whether or not Joanne and Jackie have the same birthday.
So here we must look for another approach to the problem.

A good approach can often be found by thinking the other way around:
What is the probability that it is NOT true that at least two of them 
have the same birthday? Or: what is the probability that all of them 
have different birthdays? When we have computed that, for example we 
might find that it is be 0.4 (= 40%), then the probability we look for 
is 'the rest', which in this case would be 1 - 0.4 = 0.6 (= 60%).

Okay, let's look for the actual probability that Joanne, Jackie and 
Joelle all have different birthdays:

Joanne ---> always fits = 1
Jackie ---> one of the other days = 364/365
Joelle ---> again one of the other days = 363/365

Thus the probability for three different birthdays
    = 1 x 364/365 x 363/365 = 0.99...

And the probability that at least two of them have the same birthday 
is approximatly 0.01 (1%).

Now we know how to do the problem with 30 people - the class size - 
also:

1 x 364/365 x 363/365 x ... x 336/365 = 0.29...

I think you now know how to finish this one off. And I hope you have 
learned more about probability theory, too.

You might want to take a look at the Dr. Math Frequently Asked 
Questions Web page, which explains some basics of probability theory. 
You can find it at:

  http://mathforum.org/dr.math/faq/faq.prob.intro.html   

All the best,

- Doctor Floor, The Math Forum