3. A toll facility with two automated (like E-ZPass) tollbooths A and B (in the

Question

3. A toll facility with two automated (like E-ZPass) tollbooths A and B (in the direction towards downtown) has been installed at a freeway. On any given day, the probability that tollbooth A operates properly is 0.80 and that tollbooth B operates properly is 0.90. However, the two tollbooths share some common hardware. As a result, if tollbooth A has failed the probability that tollbooth B fails at the same time is 0.40. 8 a. What is the probability that at least one lane of the to facility operates properly? b. What is the probability that both lanes operate properly?

Explanation / Answer

here let A and B are probability A and B work properly

P(B fails given A fails) =P(Bc|Ac)= P(Ac n Bc)/P(Ac) =0.4

(1-P(A U B)) =0.4*(1-P(A))

1-P(A uB) =0.4*(1-0.8)

P(A u B) =1-0.08

P(A u B)=0.92

hence probability that at least one lance of toll operate properly =0.92

b)P(both lanes operate properly)=P(A)+P(B)-P(A n B)=0.8+0.9-0.92 =0.78

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