Automobiles arrive at a vehicle equipment inspection station

Automobiles arrive at a vehicle equipment inspection station according to a Poisson process with rate alpha = 12 per hour. Suppose that with probability 0.5 an arriving vehicle will have no equipment violations.

(a) What is the probability that exactly ten arrive during the hour and all ten have no violations? (Enter your answer to five decimal places.)

(b) For any fixed y >=12, what is the probability that y arrive during the hour, of which twelve have no violations? (Include exclamation points and multiplication symbols as necessary.)

(c) What is the probability that ten \"no-violation\" cars arrive during the next hour? (Enter your answer to four decimal places.)

Solution

Mean = 12

P(violations) = P(no violations) = 0.5

a) Exactly 10 no-violation cars :

(mean)^x * e^(-mean) / x!

(12)^10 * e^(-12) / 10!

0.1048372558836593

Now, all these must have no violations

0.1048372558836593 * (0.5)^10

0.00010 ---> ANSWER to a

---------------------------------------------------------------------