Imagine
a group of 40 people. What do you think is the probability that 2 of
these people have their anniversary on a same date?
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Imagine
you just won a TV-quiz, and the quiz master gives you the chance to go
home with a car that is hidden behind one of three closed doors (A, B, and C). The
quiz master asks you to select a door (without opening it), and you select door A.
Then, the quiz master opens door C, behind which no car appears to be, and he asks you whether you want to stick to door A or
change to door B. What should you do?
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Imagine
you are a
medical doctor conducting a population research into the occurrence of
some
desease, by means of a test that gives the correct diagnose (yes or no)
in
90% of the cases. If you just tested someone as being positive, does
this person have a 90% probability of having this desease?
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Many
people would
say intuitively that this probability is about 15%, but it is
actually
more than 90%.
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Many
people would
say intuitively that your chances do not change by the quiz master's
opening of the car-less door C, and that he is just teasing you. In
fact, however, he is giving you the opportunity to double your chances.
Once he has opened door C, the probability that the car is behind door
B remains 33%, but the probability that the car is behind door A
doubles to 66%. So, change to door A!
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Many
medical doctors seem to think so, but it actually depends strongly on
how rare this desease is. If, for instance, 2% of the entire population
has this desease, then a
positive test result implies that the tested person has only a 15.5%
probability of having this desease. Therefore, as a medical doctor, you
should retest this person. If you retest this person with the same
test, yielding again a positive test result, then the probability that
this person has this desease increases to 56.5%.
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Peter
A. van der Helm 