1 Bayes Thm allows us to derive the Probability of B given A P B A as long as we know the Probability of A P A the Probability of B P B and the Probability of A given B P A B P B A P A B P B P A 2 An attempt at intuition Let s say you can stand in either Hoop A or Hoop B or both 70 of the time you stand in Hoop A BLUE or GREEN 60 of the time you stand in Hoop B YELLOW or GREEN Hoop A and Hoop B overlap so that if you are standing in Hoop B 50 of that time you are also standing in Hoop A GREEN Let s say you re standing in Hoop A What are the chances that you are also standing in Hoop B Using the RHS of Equation 2 we can solve P Hoop B Hoop A 50 60 43 70 In order to absorb the intuitiveness of this look at the paragraph above the diagram to see where these percentages come from Note that if you knew 30 of the time you were standing in both hoops GREEN you could easily calculate the LHS of Equaion 2 P Hoop B Hoop A 30 43 70 This thm is used when we need to indirectly deduce the conditional probability of an event particularly if that conditional probability cannot be directly observed It is often used with drug steroid and other medical tests However let s assume you re a fighter pilot In your F 16 you have an Identification Friend or Foe IFF system This radar system identifies other aircraft in your proximity as either friend or foe Let s say your IFF system gives the correct identification 95 of the time That is P Identify friend actual friend 0 95 P Identify foe actual foe 0 95 You are flying in a simulated environment where you know there are 80 friend and 20 foe aircraft in your proximity Your IFF displays a foe tag on an aircraft just ahead of you What s the probability that the aircraft is actually a foe P B A P Actual foe identified foe We know P A P Identified foe P Identify foe actual foe P Actual foe P Identify foe actual friend P Actual friend 0 95 0 2 0 05 0 8 0 23 P B P Actual foe 0 2 P A B P Identify foe actual foe 0 95 Using Bayes Thm P Actual foe identified foe 0 95 0 2 0 83 0 23