A service station has a pump that distributes diesel fuel to automobiles. The st

Question

A service station has a pump that distributes diesel fuel to automobiles. The station owner estimates that only about 3.2 cars use the diesel pump every 2 hours. Assume the arrivals of diesel pump users are Poisson distributed. a. What is the probability that three cars will arrive to use the diesel pump during a one-hour period? b. Suppose the owner needs to shut down the diesel pump for half an hour to make repairs. However, the owner hates to lose any business. What is the probability that no cars will arrive to use the diesel pump during a half-hour period? c. Suppose five cars arrive during a one-hour period to use the diesel pump. What is the probability of five or more cars arriving during a one-hour period to use the diesel pump? If this outcome actually occurred, what might you conclude?

Explanation / Answer

5.40(a) Average number of cars will arrive in one hour = 3.2/2 = 1.6 cars per hour

P(X = 3 cars; 1.6) by possion distribution = e-x/ x! = e-1.6 (1.6)3/ 3! = 0.1378

so there is 13.78 % probability that exactly 3 car will arrive in one hour

(b) during half an hour period, average number cars will arrive = 3.2/2 * 0.5 = 0.8 cars per hour

so P( X = 0 cars; 1.6) by possion distribution = e-x/ x! = e-0.8 (0.8)0/ 0! = 0.4493

so there is 45% chance that there will be no car aarival in that period.

c) So, here we have to calculate the probability that 5 or more cars will arive

P( X >= 5; 1.6) = by possion distribution = e- x/ x! = e-1.6(1.6)5/ 5! = 0.02368

so here we can see that there is only 2.37% chance that 5 or more cars will arrive during one hour period which is very unusual. I this outcome occured, then it is a rare event

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