3. Let P and F denote payment and default of a credit card, respectively. Suppos

Question

3. Let P and F denote payment and default of a credit card, respectively. Suppose that whether a person will pay or fail to pay the credit card in the given month depends on the past payment history only through the payment behavior in the two immediately preceding months based on the following probabilities: The probability that the person makes the payment given that he/she paid in the immediately preceding two months is Prob(PIP, P) 0.9. The probability that the person makes the payment given that he/she made the payment in the immediately preceding month and failed to pay the month before was Prob (PIF, P) 30.6. The probability that the person makes the payment given that he/she failed to pay in the immediately preceding month and paid in the month before was Prob(PIP, F) 0.4. Similarly, Prob(PIF, F) 0.1, Prob (FIP, P) 0.1, Prob(FIF, P) 0.4, Prob (FIP, F) 0.6, POFIF, F) 0.9. Let n 0 denote the initial month (month 0), let n 1 denote the 1st month, and so on. We also know the following tow initial probabilities: Prob(the person fails to pay in the initial month, and he pays 1st month) Prob(the person fails to pay in the initial and month) 0.8

Explanation / Answer

P(FP) = 0.2

P(FF) = 0.8

P(XFF) = The required probability of failing in the 2nd and 3rd months where X may be P or F.

So, the required probability is : P(PF).P(F | P, F) + P(FF).P(F | F, F) = 0.2*0.6+0.8*0.9 = 0.84

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