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
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