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Modelling the randomness of events
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| Let us say that we observe patiently a petty-crook cheating
passers by with a die: He places bets that if he throws a die and it comes
out 1, 3 or 5 the passer-by gets a pound, but if it comes up 2, 4 or 6, the
passer-by pays a pound. If we stand and watch for a while, we shall see that
out of the 20 throws, 1 came up only once, 3 twice and 5 once. 2 and 6 came
up five times each and 4 came up six times. Although we may never hold or
study closely the die this petty-crook uses, we may develop a pretty good
idea from these results that this die is loaded. We may even try to model
its behaviour by a function:
( Here Then if our "friend" asks us to play dice with him, we know that our prior probability of winning is pretty grim; in fact we may predict that it is only 20%. The same way we know that the prior probability of falling from the 10th floor of a block of flats and surviving is less than 1%. We may not know which die God is going to use if we take such a plunge, but we may pretty well predict its outcome!
Scientists like to use prior probabilities of events. They usually calculate them as frequencies of observed outcomes: If you look at the newspapers and see how many people survived after falling from the 10th floor, you can calculate the "frequency of survival" as a fraction of the number of people who survived over the number of all incidents you could find reported in the newspapers! In the same way you can predict the prior probability of a hectare of forest being burned this year if you divide the number of hectares burned last year by the total number of forested hectares in the world. |
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means integer
part of n divided by 2, i.e. it is 2 if n=5 or n=4, it is 0 if
n=1, etc.)
