Advanced Probability Topics Probability 1 Law of Total

Advanced Probability Topics Probability 1

Law of Total Probability � Suppose we don’t know the probability of a specific event, but we know how it related to another. � For any two events A and B: � Applying the multiplication rule: 2

Total Probability w/ Multiple events � Assume events. E 1, E 2, … Ek are k mutually exclusive subsets of 3

Bayes’ Theorem � Often we observe conditional probabilities through prior information � Notes: ◦ Found by rewriting the multiplication rule ◦ Reversal of the condition in numerator ◦ Whatever you are “given” in the Denominator �Often can use LTP to find denominator 4

Bayes Theorem + Law of Total Probability Extended E 1, E 2, … Ek are k mutually exclusive and exhaustive events and B is any event, � If � Note: ◦ Total probability expression of the denominator ◦ Numerator is always a term of the denominator. 5

Law of Total Probability Example Information about product failure based on chip manufacturing process contamination. � � ◦ Level of Contamination Probability of Level Probability of Failure, given Level High (H) P(H) = 0. 2 P(F|H) = 0. 1 Not High (H’) P(H’) = 0. 8 P(F|H’) = 0. 005 Find the overall Probability of Failure: To find P(F) use the law of total probability: � � P(F) = P(F ∩ H) + P(F ∩ H’) P(H) = 0. 2 and P(F|H) = 0. 100, so P(F ∩ H) = 0. 020 P(H’) = 0. 8 and P(F|H’) = 0. 005, so P(F ∩ H’) = 0. 004 P(F) = P(F ∩ H) + P(F ∩ H’) = 0. 020 + 0. 004 = 0. 024 6

Bayes’ Theorem Example � � Recall the previous example: Level of Contamination Probability of Level Probability of Failure, given Level High (H) P(H) = 0. 2 P(F|H) = 0. 1 Not High (H’) P(H’) = 0. 8 P(F|H’) = 0. 005 Now suppose we have a chip that failed, and we want to know the probability it had a high level of contamination. � We know from the previous problem using the LTP P(F) = 0. 024 � Interpretation: If it failed, there is an 83% chance the contamination level was high 7

Bayesian Network Example printer manufacturer determines it’s Printer failures (F) are of 3 types. They also know how likely the Printers are to fail given the type of problem you are experiencing. �A Type of Problem Probability of Failure Given problem Hardware P(H) = 0. 1 P(F|H) = 0. 9 Software P(S) = 0. 6 P(F|S) = 0. 2 Other P(O) = 0. 3 P(F|O) = 0. 5 the max of P(H|F), P(S|F), P(O|F) to direct the diagnostic effort. � Find 8

Bayesian Network Example Solved � First we need to overall probability of failure: � This will be used in the denominator for each of our “pieces” � Our most likely scenario is an “other” issue given failure 9