Question

Your car was in a small accident and you took it to Able Autobody first thing on a Monday morning. Able services cars with rate λ =1/2 /hour,

i.e... P(Current Car Finished in [t,t+τ])≈τ/2

for small τ, and works 8 hours a day (not including lunch). You are third in line including the car being worked on when the shop opens.

Find the probability that your car will not be fixed that day.

Answer 1

Let Ti be the service time of the ith car.

Assuming, Service time Ti ~ Exponential( = 1/2)

If you are third in line, then your car will not be fixed that day if T1 + T2 + T3 > 8 hours

We know that the sum of exponential distribution follows Gamma Distribution.

T = T1 + T2 + T3 ~ Gamma(n = 3, = 1/2)

Probability that your car will not be fixed that day = P(T > 8)

= 1 - P(T < 8)

Now, CDF of Gamma(, ) is,

P(T < 8) =

Probability that your car will not be fixed that day = P(T > 8)

= 1 - P(T < 8) = 1 - 0.7710545 = 0.2289455