Exercise 1: Fraud Detection With Bayes' Theorem — Possible Solution
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GIVEN
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P(flagged | fraud)     = 0.95
P(flagged | not fraud) = 0.02
P(fraud)               = 0.001  (0.1% of all transactions)

STEP 1: THE LAW OF TOTAL PROBABILITY — P(flagged)
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P(flagged) = P(flagged|fraud) x P(fraud) + P(flagged|not fraud) x P(not fraud)
           = (0.95 x 0.001) + (0.02 x 0.999)
           = 0.00095 + 0.01998
           = 0.02093

STEP 2: APPLYING BAYES' THEOREM
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P(fraud|flagged) = [P(flagged|fraud) x P(fraud)] / P(flagged)
                  = 0.00095 / 0.02093
                  ~= 0.0454 (about 4.54%)

INTERPRETATION
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Even though the system correctly flags 95% of real fraud, a flagged
transaction is only actually fraudulent about 4.5% of the time. This
is this chapter's own base-rate-neglect effect: because genuine fraud
is so rare (0.1% of all transactions), the much larger pool of
legitimate transactions produces far more false positives (2% of a
huge group) than the fraud detector produces true positives (95% of
a tiny group).

WHY THIS WORKS AS AN ANSWER
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The total probability of a flag is built first using this chapter's
own law of total probability (accounting for both the fraud and non-
fraud paths to a flag), then Bayes' Theorem is applied directly to
convert the known P(flagged|fraud) into the actually useful
P(fraud|flagged), matching the exact two-step method demonstrated in
this chapter's own worked examples.