b. Lyman Ott Michael Longnecker

This definition for conditional probabilities gives rise to what is referred to as the multiplication law. Suppose that you have the additional information that the claim was associ- ated with a fire policy. Checking Table 4.2, we see that 20 or .20 of all claims are associated with a fire policy and that 6 or .06 of all claims are fraudulent fire policy claims. Therefore, it follows that the probability that the claim is fraudulent, given that you know the policy is a fire policy, is This probability, PF 兩fire policy, is called a conditional probability of the event F—that is, the probability of event F given the fact that the event “fire policy” has already occurred. This tells you that 30 of all fire policy claims are fraudulent. The vertical bar in the expression PF 兩fire policy represents the phrase “given that,” or simply “given.” Thus, the expression is read, “the probability of the event F given the event fire policy.” The probability PF ⫽ .10, called the unconditional or marginal probability of the event F, gives the proportion of times a claim is fraudulent—that is, the pro- portion of times event F occurs in a very large infinitely large number of repeti- tions of the experiment receiving an insurance claim and determining whether the claim is fraudulent. In contrast, the conditional probability of F, given that the claim is for a fire policy, PF 兩fire policy, gives the proportion of fire policy claims that are fraudulent. Clearly, the conditional probabilities of F, given the types of policies, will be of much greater assistance in measuring the risk of fraud than the unconditional probability of F. ⫽ .06 .20 ⫽ .30 PF 冷 fire policy ⫽ proportion of claims that are fraudulent fire policy claims proportion of claims that are against fire policies TABLE 4.2 Categorization of insurance claims Type of Policy Category Fire Auto Other Total Fraudulent 6 1 3 10 Nonfraudulent

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47 90 Total 20 30 50 100 conditional probability unconditional probability DEFINITION 4.7 Consider two events A and B with nonzero probabilities, PA and PB. The conditional probability of event A given event B is The conditional probability of event B given event A is PB 冷 A ⫽ PA 傽 B PA PA 冷 B ⫽ PA 傽 B PB DEFINITION 4.8 The probability of the intersection of two events A and B is ⫽ PBPA 冷 B PA 傽 B ⫽ PAPB 冷 A