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?

Answer 1

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