Cheers whufc65, i still got over 5 hours of hard w**king to do and your already teasing me with naughty pictures!!
no trick. with 23 people the probability of 2 of them sharing a birthday is 50.73%
How do you come to a percentage figure of 50.73%? 23 people and a possible 366 dates that still seems too high to me
Obviously not in my own words.........
To solve the birthday problem, we need to use one of the basic rules of probability: the sum of the probability that an event will happen and the probability that the event won't happen is always 1. (In other words, the chance that anything might or might not happen is always 100%.) If we can work out the probability that no two people will have the same birthday, we can use this rule to find the probability that two people will share a birthday:
P(event happens) + P(event doesn't happen) = 1
P(two people share birthday) + P(no two people share birthday) = 1
P(two people share birthday) = 1 - P(no two people share birthday).
So, what is the probability that no two people will share a birthday?
Again, the first person can have any birthday. The second person's birthday has to be different. There are 364 different days to choose from, so the chance that two people have different birthdays is 364/365. That leaves 363 birthdays out of 365 open for the third person.
To find the probability that both the second person and the third person will have different birthdays, we have to multiply:
(365/365) * (364/365) * (363/365) = 132 132/133 225,
which is about 99.18%.
If we want to know the probability that four people will all have different birthdays, we multiply again:
(364/365) * (363/365) * (362/365) = 47 831 784/ 48 627 125,
or about 98.36%.
We can keep on going the same way as long as we want. A formula for the probability that n people have different birthdays is
((365-1)/365) * ((365-2)/365) * ((365-3)/365) * . . . * ((365-n+1)/365).
If you know permutation notation, you can write this formula as
(365_P_n)/(365^n).
That's the same as
365! / ((365-n)! * 365^n).
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We've made some progress, but we still haven't answered the original question: how large must a class be to make the probability of finding two people with the same birthday at least 50%?
We know that the probability of finding at least two people with the same birthday is 1 minus the probability that everybody has a different birthday, and we know how to find the probability that everybody has a different birthday for any number of people. The easiest way to find the right class size is to use a calculator to try different numbers in the formula. It turns out that the smallest class where the chance of finding two people with the same birthday is more than 50% is... a class of 23 people. (The probability is about 50.73%.)
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