Discrete Mathematics University of Kentucky CS 275 Spring ... - MGNet

Example: You receive n items. Sometimes all n items are guaranteed to be good.
However, not all shipments have been checked. The probability that an item is
bad in an unchecked batch is 0.1. We want to determine whether or not a
shipment has been checked, but are not willing to check all items. So we test
items at random until we find a bad item or the probability that a shipment
seems to have been checked is 0.001. How items do we need to check? The
probability that an item is good, but comes from an unchecked batch is 140.1 =
0.9. Hence, the k th check without finding a bad item, the probability that the
items comes from an unchecked shipment is (0.9) k . Since (0.9) 66 ~0.001, we must
check only 66 items per shipment.
Theorem: If the probability that an element of a set S does have a particular
property is in (0,1), then there exists an element in S with this property.
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Bayes Theorem: Suppose that E and F are events from a sample space S such
that p(E) - 0 and p(F) - 0. Then
p(F|E) = p(E|F)p(F) / (p(E|F)p(F) + p(E|F)p(F)).
Generalized Bayes Theorem: Suppose that E is an event from a sample space
n
and that F 1 , F 2 , …, F n are mutually exclusive events such that ! F i
= S.
i=1
Assume that p(E) - 0 and p(F i ) - 0, 1;i;n. Then
n
i=1
p(F j |E) = p(E| F j )p(F j ) / ! p(E|F i
)p(F i
).
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