Drivers arrive at a toll booth at a rate of 3 per minute during peak traffic periods.The time between consecutive driver arrivals follows an exponential distribution.What is the probability that it will take more than 1/3 of a minute between consecutive drivers?
A)0.632
B)0.111
C)0.368
D)0.208
E)Not enough information given

Question

Quizplus · Accepted Answer

The time between consecutive driver arrivals follows an exponential distribution. 
Let X be the time between consecutive driver arrivals. Then X ~ Exp(3) (since the rate is 3 per minute).

We need to find P(X > 1/3).

Using the cumulative distribution function (CDF) of the exponential distribution, we have:

P(X > 1/3) = 1 - P(X ≤ 1/3) 
= 1 - F(1/3), where F is the CDF of X

The CDF of X is given by:

F(x) = 1 - e^(-λx), where λ is the rate parameter.

Substituting λ = 3 and x = 1/3, we get:

F(1/3) = 1 - e^(-3/3) = 1 - e^(-1)

Therefore,

P(X > 1/3) = 1 - F(1/3) = e^(-1) ≈ 0.368

Hence, the probability that it will take more than 1/3 of a minute between consecutive drivers is 0.368, or approximately 0.37.

Therefore, the answer is (C).

Drivers Arrive at a Toll Booth at a Rate of 3