Collina’s Italian Café in Houston, Texas, advertises that carryout orders take about 25 minutes (Collina’s website, February 27, 2008). Assume that the time required for a carryout order to be ready for customer pickup has an exponential distribution with a mean of 25 minutes.
What is the probability than a carryout order will be ready within 20 minutes (to 4 decimals)?
If a customer arrives 30 minutes after placing an order, what is the probability that the order will not be ready (to 4 decimals)?
A particular customer lives 15 minutes from Collina’s Italian Café. If the customer places a telephone order at 5:20 P.M., what is the probability that the customer can drive to the café, pick up the order, and return home by 6:00 P.M. (to 4 decimals)?
a) .5507
b) .3012
To answer these questions, we can use the exponential distribution with a mean of 25 minutes. The probability density function (pdf) of the exponential distribution is given by:
f(x) = (1/μ) * exp(-x/μ)
where x is the time (minutes), and μ is the mean (25 minutes in this case).
1. Probability that a carryout order will be ready within 20 minutes:
To find this probability, we need to calculate the cumulative distribution function (CDF) at 20 minutes.
F(x) = 1 - exp(-x/μ)
Using the CDF formula and plugging in the values, we get:
F(20) = 1 - exp(-20/25)
Therefore, the probability that a carryout order will be ready within 20 minutes is approximately 0.4866 (to 4 decimals).
2. Probability that the order will not be ready after 30 minutes:
Again, we can use the CDF to find this probability. Since the CDF gives the probability that the order will be ready before a given time, the probability that the order will not be ready is given by:
P(Order not ready) = 1 - F(x)
Using the CDF formula and plugging in the values, we get:
P(Order not ready) = 1 - (1 - exp(-30/25))
Therefore, the probability that the order will not be ready after 30 minutes is approximately 0.2231 (to 4 decimals).
3. Probability that the customer can pick up and return home within 40 minutes:
To calculate this probability, we need to consider both the time to drive to the café and the time for the order to be ready. This involves finding the joint probability of both events.
P(Time to pick up and return within 40 minutes) = P(Time to drive to café ≤ 40 - Time for order to be ready)
Using the exponential distribution, we find:
P(Time to drive to café ≤ 40 - Time for order to be ready) = ∫[0,40] f(x) * F(40 - x) dx
Using this formula and integrating, we can find the probability. However, this calculation requires numerical integration which is beyond the scope of this explanation. You would need to use a mathematical software or a statistical calculator to evaluate the integral and find the probability.
Therefore, the exact probability that the customer can pick up and return home within 40 minutes cannot be provided without performing the numerical integration.