Section 4.6 Probability
309
P(F 0 E 2 ) =
P(E 2 d F )
P(E 2 )
or P(E 2 d F ) = P(F 0 E 2 )P(E 2 )
Because F d E 2 = E 2 d F, P(F d E 2 ) = P(E 2 d F) and therefore
P(F d E 2 ) = P(F 0 E 2 )P(E 2 ) [and a similar equation for P(F d E 1 )]
and
P(E 2 0 F) =
P(F d E 2 )
P(F)
=
P(F 0 E 2 )P(E 2 )
P(F)
F is another event in (that is, subset of) S, and
F = F d S = F d (E 1 c E 2 ) = (F d E 1 ) c (F d E 2 )
so F is the union of disjoint events, and
P(F ) = P(F d E 1 ) + P(F d E 2 ) = P(F 0 E 1 )P(E 1 )+ P(F 0 E 2 )P(E 2 )
=
43
100
# 1
2
+
61
100
# 1
2
=
43 + 61
200
=
104
200
Finally,
P(E 2 0 F ) =
P(F 0 E 2 )P(E 2 )
P(F )
=
(61∙100)(1∙2)
104∙200
=
61∙200
104∙200
= 61∙104 > 0.587
As we suspected, the probability that the spinach came from the contaminated
batch is greater than 0.5.
The general statement of Bayes’ theorem (see Exercise 89) follows.
tHeoRem BayeS’ theORem
Let E 1 , … , E n be disjoint events from a sample space S whose union equals S. If
F is another event from S, then the probability of event E i , 1 ≤ i ≤ n, given event
F, is
P(E i 0 F) =
P(F 0 E i )P(E i )
∙
n
k=1
P(F 0 E k )P(E k )
PraCtiCe 46 In Example 73, what is the probability that the package came from Supplier A if it was
lettuce?
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