# Law of total probability

Let A_{1}, A_{2},…,A_{n} events. They form a **complete system of events** if:

–A_{1}UA_{2}U…UA_{n} = E

–they are independent in pairs (A_{i}∩A_{j} = ф, i,j = 1, 2,…,n)

**Law of total probability**: Let A_{1}, A_{2},…,A_{n} a complete system of events with P(A_{i}) ≠ 0, i = 1, 2,…,n. Let B another event which we know P(B/A_{i}), 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

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