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

Question

Quizplus · Accepted Answer

The rate parameter λ of the exponential distribution is equal to the average rate of arrival, which is 3 drivers per minute. Therefore, λ = 3. 
The probability that it will take more than 1/3 of a minute between consecutive drivers is the same as the probability of waiting more than 1/3 of a minute for the next arrival, which is P(X > 1/3), where X is the time between consecutive arrivals. 
This probability can be calculated using the cumulative distribution function (CDF) of the exponential distribution: 
P(X > 1/3) = 1 - F(1/3) 
where F(t) = 1 - e^(-λt) is the CDF of exponential distribution. 
Substituting λ = 3 and t = 1/3, we get: 
P(X > 1/3) = 1 - (1 - e^(-1)) ≈ 0.368 
Therefore, the answer is (C) 0.368.

Drivers Arrive at a Toll Booth at a Rate of 3