find the revised probability that it is causeu by polu 45 A factory has three pr

Question

find the revised probability that it is causeu by polu 45 A factory has three production lines 1, 2, and 3 contributing 20%, 30%, and 50%, respectively, to its total output. The percentages of substan- dard items produced by lines 1, 2, and 3 are, respectively, 15, 10, and 2. (a) If an item is chosen at random from the total output, what is the probability that it will be substandard? (b) Suppose that an item chosen at random from the total output is found to be substandard. What is the probability that the item is from line 1?

Explanation / Answer

Let Event S: Item is standard
Let Event L1: Item is from Production line 1
Let Event L2: Item is from Production line 2
Let Event L3: Item is from Production line 3
Given: P(L1) = 0.20, P(L2) = 0.30, P(L3) = 0.50
P(S | L1) = 0.15, P(S | L2) = 0.10, P(S | L3) = 0.02
(a)
P(S), Using total law of probability
P(S) = P(L1)P(S | L1) + P(L2)P(S | L2) + P(L3)P(S | L3)
P(S) = (0.20 x 0.15) + (0.30 x 0.10) + (0.50 x 0.02) = 0.07
(b)
Probability that the item is from production line 1 given that it is a standard item: P(L1 | S), Using conditional probability formula, we can write
P(L1 | S) = P(L1 S) / P(S) = P(L1)P(S | L1) / P(S)
P(L1 | S) = (0.20 x 0.15) / 0.07 = 0.4286

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