10. Bayes' Theorem
Let E1 and E2 be two mutually exclusive events forming a partition of the sample space S and let E be any event of the sample space such that P(E) ≠ 0.
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Example
The sample space S is described as "the integers 1 to 15" and is partitioned into:
E1 = "the integers 1 to 8" and
E2 = "the integers 9 to 15".
If E is the event "even number" then we have the following:
[Recall from Conditional Probability that the notation P(E1 | E) means "the probability of the event E1 given that E has already occurred".]
Statement of Bayes' Theorem
The probabilities for the situation described above is given by Bayes' Theorem, which can be calculated in two ways:
So for our example above, checking both items of this equation:
We get the same result using the second form:
Extending Bayes' Theorem for Mutually Exclusive Events
Bayes' Theorem can be extended as follows:
If E1, E2, ... , Ek are mutually exclusive events forming partitions of the sample space S and if E is any event of S such that P(E) ≠ 0, then
EXAMPLE 1
Of all the smokers in a particular district, 40% prefer brand A and 60% prefer brand B. Of those smokers who prefer brand A, 30% are females, and of those who prefer brand B, 40% are female. What is the probability that a randomly selected smoker prefers brand A, given that the person selected is a female?
EXAMPLE 2
There are 3 urns A, B and C each containing a total of 10 marbles of which 2, 4 and 8 respectively are red. A pack of cards is cut and a marble is taken from one of the urns depending on the suit shown - a black suit indicating urn A, a diamond urn B, and a heart urn C. What is the probability a red marble is drawn?
If somebody secretly cut the cards and drew out a marble and then announced to us a red marble had in fact been drawn, could we compute the probability of the cut being, say, a heart (or more generally, can we compute the probability of a specified prior event given that the subsequent event did occur)?
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