Question

1. Customers arrive at the drive-up window of a fast-food restaurant at a rate of 2 per minute during the lunch hour
   1. What is the probability that the next customer will arrive within 1 minute?
   2. What is the probability that the next customer will arrive after 0.5 minutes?
   3. What is the probability that the next customer will arrive between 45 seconds and 1.5 minutes?

Answer 1

t : time between arrival

: Average number of arrivals

t follows a exponential distribution

Probability density function: pdf

And Cumulative distribution function CDF

Given,

Customers arrive at the drive-up window of a fast-food restaurant at a rate of 2 per minute

i.e = 2

a. probability that the next customer will arrive within 1 minute = p(T<1) = F(1)

probability that the next customer will arrive within 1 minute = 0.8647

b. probability that the next customer will arrive after 0.5 minutes = p(T>-.5) = 1-P(T<0.5) = 1-F(0.5)

p(T>-.5) = 1-P(T<0.5) = 1-F(0.5) = 1-0.6321=0.3679

probability that the next customer will arrive after 0.5 minutes = 0.3679

c. probability that the next customer will arrive between 45 seconds and 1.5 minutes

45 seconds = 45/60=0.75 minutes

probability that the next customer will arrive between 45 seconds and 1.5 minutes = P(0.75 < T <1.5)

P(0.75 < T <1.5) = P(T<1.5) - P(T<0.75) = F(1.5) - F(0.75)

P(0.75 < T <1.5) = P(T<1.5) - P(T<0.75) = F(1.5) - F(0.75) = 0.9502 - 0.7769 = 0.1733

probability that the next customer will arrive between 45 seconds and 1.5 minutes = 0.1733