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June 23, 2006

Statistics for beginners

Although I like mathematics quite well I've never been really fond of statistics. It wasn't until last week when a colleague introduced me to the "Goat Problem" that I realised how much fun it can be.

The story is about a quiz show where in the last round the candidate is given a choice of three doors. Behind two of them is a goat, behind the last one a luxury car. When the candidate has chosen a door the quiz master says: "I'll to show you something" and opens one of the other two behind which he knows is a goat. "In view of this would you like to stay with your first choice of doors or change to the other one?"

The question is what is the right strategy to maximise your chances: change or not? Change, of course! Because your chance of choosing a goat door at first and therefore getting the car by changing doors after the quiz master has opened the door with the other goat behind it is twice as high as choosing the car door at the first attempt and consequently loosing when changing doors after the quiz master has opened another door.

The real fun is when you put this problem to other people and they vehemently deny that your chances increase by changing doors even stating that after one of the doors has been opened by the quiz master you have only a 50 percent chance of getting the car, which is nonsense, of course. You can easily prove your point by offering people a choice of coins or beermats.

In 1991 one of the readers of the US magazine "Parade" asked Marylin vos Savant who answered readers' questions under "Ask Marylin" about the goat problem. Marylin said that of course he should change doors after the quiz master had opened the one with the goat behind it. That advice caused a flood of letters to the editor. 92 % of the roughly 10,000 writers - among them an astonishing number of mathematicians - contradicted Marylin in often quite rude terms. They called it a "national crisis in mathematical education". One of them even called Marylin the only goat in the whole story. 15 years later I find it hard to believe that people could react like that when a little quiet contemplation would have shown that Marylin was quite right.

Another amusing little exercise is to calculate the probability that in a group of k people two are born on the same day. For a group of 10 people the probability is 11.7%, for 50 it is 97% and for a group of 100 people the probability is 99.99996% Interestingly you need to meet a much larger group to find someone who is born on the same day you are. If you are part of a group of 500 there's only a 74.6% chance of finding someone who is born on the same day and you need to meet 4999 to raise the probability to 99.9999%! Mind you, that's only theory. At school there were three of us in a class of 24 who were born on the same day ...

For your amusement the calculations are given below..

Two persons born on the same day

For simplicity we disregard February 29. And the problem is easier to solve if we try to find the probability that no two persons are born on the same day. If the first person is born on any of the 365 days of a year, there are only 364 days left for the second persons, only 363 days for the third person et cetera. The probability for these events to occur at the same time are calculated my multiplying the probabilities:

(364/365) x (363/365) x (362/365) x ... x ((364-k+2)/365)

If we now want to know the probability of two people being indeed born on the same day we just have to subtract the result from 1.

Another person born on the same day as you:

The formula for this is much simpler because the day of birth is fixed in this case:

1 - (364/365)k

Posted by Mausi at June 23, 2006 06:21 PM

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I love this stuff. Thanks.

Posted by: vw bug at June 23, 2006 12:14 PM

Thank you, vw bug! I knew I could count on you to enjoy this.

Posted by: Mausi at June 23, 2006 03:50 PM