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Question
A, B and C was 50%, 30% and 20% of the cars in a service station respectively. They fail to clean the glass in 5%, 7% and 3% of the cars respectively. The glass of a washed car is checked. What is the probability that the glass has been cleaned?
Solution
Let E1, E2, E3 and A be the events defined as follows:
E1 = Cars in the service station A
E2 = Cars in the service station B
E3 = Cars in the service station C
A = Event of failing to clean the glass
P`("E"_1) = 50/100`, P`("E"_2) = 30/100`, P`("E"_3) = 20/100`
P`("A"/"E"_1) = 5/100`, P`("A"/"E"_2) = 7/100`, P`("A"/"E"_3) = 3/100`
Using the law of total probability
P(A) = P(failing to clean the glass)
= `"P"("E"_1) xx "P"("A"/"E"_1) + "P"("E"_2) xx "P"("A"/"E"_2) xx "P"("E"_3) xx "P"("A"/"E"_3)`
= `50/100 xx 5/100 + 30/100 xx 7/100 + 20/100 xx 3/100`
= `250/(10,000) + 210/(10,000) + 60/(10,000)`
= `520/(10,000)`
P(A) = 0.052
∴ P(that the glass has been clean)
= 1 − 0.052
= 0.948
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