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Law of total probability

Let A1, A2,…,An events. They form a complete system of events if:

–A1UA2U…UAn = E

–they are independent in pairs (Ai∩Aj = ф, i,j = 1, 2,…,n)

Law of total probability: Let A1, A2,…,An a complete system of events with P(Ai) ≠ 0, i = 1, 2,…,n. Let B another event which we know P(B/Ai), i = 1, 2,…,n. Then:


Example: A transport company has three routes within a county. 60% of its buses cover the first route, 30% the second and 10% the third. It is known that the daily probabilities of a breakdown in each route are 2%, 4% and 1%, respectively. Find out the probability for a bus to have a breakdown any day.



Exercise: A company produces two types of car parts: A and B. 20% of parts are type A and 80% are type B. The probability that type A part is defective is 0.02 and the probability that a type B part is defective is 0.1. If we choose a part randomly, what is the probability that it is defective?








Solution: 0.084