68 Chapter 3:Elements of Probability
EFEFFIGURE 3.5 P(E|F)=PP(EF(F)).
probability ofEgiven thatFhas occurred, and is denoted by
P(E|F)A general formula forP(E|F) that is valid for all eventsEandFis derived in the same
manner as just described. Namely, if the eventFoccurs, then in order forEto occur it is
necessary that the actual occurrence be a point in bothEandF; that is, it must be inEF.
Now, since we know thatFhas occurred, it follows thatFbecomes our new (reduced)
sample space and hence the probability that the eventEFoccurs will equal the probability
ofEFrelative to the probability ofF. That is,
P(E|F)=P(EF)
P(F)(3.6.1)Note that Equation 3.6.1 is well defined only whenP(F)>0 and henceP(E|F) is defined
only whenP(F)>0. (See Figure 3.5.)
The definition of conditional probability given by Equation 3.6.1 is consistent with the
interpretation of probability as being a long-run relative frequency. To see this, suppose
that a large numbernof repetitions of the experiment are performed. Then, sinceP(F)
is the long-run proportion of experiments in whichFoccurs, it follows thatFwill occur
approximatelynP(F) times. Similarly, in approximatelynP(EF) of these experiments,
bothEandFwill occur. Hence, of the approximatelynP(F) experiments whose outcome
is inF, approximatelynP(EF) of them will also have their outcome inE. That is, for
those experiments whose outcome is inF, the proportion whose outcome is also inEis
approximately
nP(EF)
nP(F)=P(EF)
P(F)Since this approximation becomes exact asnbecomes larger and larger, it follows that
(3.6.1) gives the appropriate definition of the conditional probability ofEgiven thatFhas
occurred.
EXAMPLE 3.6a A bin contains 5 defective (that immediately fail when put in use), 10
partially defective (that fail after a couple of hours of use), and 25 acceptable transistors.
A transistor is chosen at random from the bin and put into use. If it does not immediately
fail, what is the probability it is acceptable?