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Sum Rule and Bayes' Theorem
  • P(B) = P(A∩B)+ P(Ac ∩B)

          = P(B│A)*P(A) + P(B│Ac)*P(Ac)
 

  • P(A|B) = P(B│A)*P(A) / [P(B│A)*P(A)+P(B│Ac)*P(Ac)]
     

Sample Question
 
Example

The subsidiary will default if the parent defaults, but the parent will not necessarily default if the subsidiary defaults. Calculate Prob. of a subsidiary & parent both defaulting. Parent has a PD = 0.5% subsidiary has PD of.9%.

Solution

P(P∩S) = P(S/P)*P(P)
= 1*0.5%
= 0.5%

 





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