The most important new idea about probability is the probability that something happens, on condition that something else happens. This is called conditional probability.
CATEGORICAL AND CONDITIONAL
We express probabilities in numbers. Here is a story I read in the newspaper. The old tennis pro Ivan was discussing the probability that the rising young star Stefan would beat the established player Boris in the semifinals. Ivan was set to play Pete in the other semifinals match. He said,
The probability that Stefan will beat Boris is 40%.
Or he could have said,
The chance of Stefan's winning is 0.4.
These are categorical statements, no ifs and buts about them. Ivan might also have this opinion:
Of course I'm going to win my semifinal match, but if I were to lose, then Stefan would not be so scared of meeting me in the finals, and he would play better; there would then be a 50–50 chance that Stefan would beat Boris.
This is the probability of Stefan's winning in his semifinal match, conditional on Ivan losing the other semifinal. We call it the conditional probability. Here are other examples:
Categorical: The probability that there will be a bumper grain crop on the prairies next summer.
Conditional: The probability that there will be a bumper grain crop next summer, given that there has been very heavy snowfall the previous winter. […]
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