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.
a)What is the probability than a carryout order will be ready within 20 minutes (to 4 decimals)?
b)If a customer arrives 30 minutes after placing an order, what is the probability that the order will not be ready (to 4 decimals)?
c)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)?
To solve these problems, we will use the exponential distribution formula. The exponential distribution is often used to model the time it takes for an event to occur. Let's calculate the probabilities step by step.
a) Probability that a carryout order will be ready within 20 minutes:
To find the probability that the order will be ready within 20 minutes, we need to calculate the cumulative distribution function (CDF) of the exponential distribution. The CDF represents the probability that the random variable X takes on a value less than or equal to a given value x.
The formula for the CDF of the exponential distribution is:
CDF(x) = 1 - e^(-λx)
Given that the mean of the exponential distribution is 25 minutes, we can find the rate parameter (λ) using the formula:
λ = 1 / mean
So, λ = 1 / 25 = 0.04
Now, we can plug in the values in the CDF formula:
CDF(20) = 1 - e^(-0.04 * 20)
Calculating this, we find that CDF(20) ≈ 0.3297
Therefore, the probability that a carryout order will be ready within 20 minutes is approximately 0.3297.
b) Probability that the order will not be ready after 30 minutes:
To find the probability that the order will not be ready after 30 minutes, we need to calculate the survival function (also called the complementary cumulative distribution function). The survival function represents the probability that the random variable X takes on a value greater than a given value x.
The formula for the survival function of the exponential distribution is:
SF(x) = e^(-λx)
Using the same rate parameter (λ = 0.04), we can find the probability as follows:
SF(30) = e^(-0.04 * 30)
Calculating this, we find that SF(30) ≈ 0.4493
Therefore, the probability that the order will not be ready after 30 minutes is approximately 0.4493.
c) Probability that the customer can pick up the order and return home within 40 minutes:
In this case, we need to calculate the probability that the carryout order will be ready within 20 minutes, which we calculated in part (a), and also consider the time required for the customer to drive to the café and return home.
Since the customer lives 15 minutes from the café, the driving time is 15 minutes one-way. Therefore, the total time required is 20 minutes (order ready time) + 15 minutes (driving time) + 15 minutes (return driving time) = 50 minutes.
To calculate this probability, we need to find the survival function at 50 minutes:
SF(50) = e^(-0.04 * 50)
Calculating this, we find that SF(50) ≈ 0.0821
Therefore, the probability that the customer can pick up the order and return home within 40 minutes is approximately 0.0821.