Unformatted text preview: 2.3 12 +22 +32 +42 +52 +62
6 = 91
6. So Conditional probabilities Let A and B be two events for some probability experiment. The conditional probability of B given
A is deﬁned by
P (AB )
if P (A) > 0
P ( A)
P (B |A) =
undeﬁned if P (A) = 0.
It is not deﬁned if P (A) = 0, which has the following meaning. If you were to write a computer
routine to compute P (B |A) and the inputs are P (AB ) = 0 and P (A) = 0, your routine shouldn’t
simply return the value zero. Rather, your routine should generate an error message such as “input
error–conditioning on event of probability zero.” Such an error message would help you or others
ﬁnd errors in larger computer programs which use the routine.
Intuitively, if you know that event A is true for a probability experiment, then you know that
the outcome ω of the probability experiment is in A. Conditioned on that, whether B is also true
should only depend on the outcomes in B that are also in A, which is the set AB. If B is equal
to A, then, given A is true,...
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