This link has been bookmarked by 2 people . It was first bookmarked on 10 Aug 2006, by Kinaydjin.
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27 Feb 09
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Often when you work out the probability of an event, you sometimes do not need to work out the probability of an event occurring, in fact you need the opposite, the probability that the event will not occur.
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The question is:-
"How many people should be gathered in a room together before it is more likely than not that two of them share the same birthday?"
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Ignoring the issues of leap years the problem is solved as follows:-
When the first person enters the room and announces their birthday, the probability of the second person sharing the same birthday is 1/365. Conversely, the probability of the second birthday being different is the opposite of the first calculation, 364/365. When two birthdays are known, the probability of the third being different is 363/365, as there are now two 'favourable' outcomes among 365. The compound probability of birthday 2 being different from birthday 1, and of birthday 3 being different from the other two, these being independent outcomes, is:-
(364/365)*(363/365) = 0.991796 or 99.2% chance that two people will not share the same birthday.
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All that is necessary now is to continue adding terms to the fraction until it equals less than 1/2 or 50%, since as soon as the probability is less than 1/2 that all birthdays are different, the probability is clearly more than 1/2 that any two are the same. In other words it is more likely than not that two people in the room share the same birthday.
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The fraction drops to less than 1/2 with 23 iterations, so it is more likely than not that in any gathering of 23 or more persons, two of them will share a birthday.
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10 Aug 06
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