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)?
To answer these probability questions, we will use the exponential distribution with a mean of 25 minutes. The exponential distribution is commonly used to model the time between events that occur at a constant average rate.
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) of the exponential distribution at x = 20 minutes. The CDF of an exponential distribution with mean λ is given by the formula:
CDF(x) = 1 - e^(-λx)
In this case, the mean is 25 minutes, so λ = 1/25. Plugging in the values, we get:
CDF(20) = 1 - e^(-20/25) ≈ 0.5768 (rounded to 4 decimals)
Therefore, the probability that a carryout order will be ready within 20 minutes is approximately 0.5768.
2. Probability that the order will not be ready after 30 minutes:
Here, we need to calculate the complement of the probability that the order will be ready within 30 minutes, since the question is asking for the probability that the order will NOT be ready. So, we can find this probability by subtracting the CDF at x = 30 minutes from 1.
Probability = 1 - CDF(30)
Using the same formula as above with λ = 1/25:
Probability = 1 - (1 - e^(-30/25)) ≈ 0.2865 (rounded to 4 decimals)
Therefore, the probability that the order will not be ready after 30 minutes is approximately 0.2865.
3. Probability that the customer can complete the entire process within 40 minutes:
To calculate this probability, we need to find the sum of the probabilities that the order will be ready at or before the customer arrives at the café and the time it takes for the customer to complete the round trip is less than or equal to 40 minutes.
First, let's calculate the probability that the order will be ready by the time the customer arrives. We can use the CDF at x = 30 minutes, which is the time it takes for the customer to get to the café.
Probability_order_ready = CDF(30)
Next, we need to calculate the probability that the round trip time for the customer is less than or equal to 40 minutes. Since the time to drive to and from the café follows the same exponential distribution with mean 15 minutes (half of the total round trip), we can use the CDF at x = 20 minutes to calculate this probability.
Probability_round_trip = CDF(20)
To find the overall probability, we multiply the two probabilities:
Probability = Probability_order_ready * Probability_round_trip
Using the exponential distribution formula with λ = 1/15 for the round trip:
Probability = CDF(30) * CDF(20)
Calculating this using the formulas mentioned earlier, we find:
Probability ≈ 0.3334 (rounded to 4 decimals)
Therefore, the probability that the customer can drive to the café, pick up the order, and return home within 40 minutes is approximately 0.3334.
To answer these questions, we will make use of the exponential distribution formula:
1. Probability of a carryout order being ready within 20 minutes:
Given that the mean of the exponential distribution is 25 minutes, we can use the cumulative distribution function (CDF) to find the probability.
P(X ≤ 20) = 1 - e^(-λx)
Where λ is the rate parameter, which is the reciprocal of the mean (1/25).
P(X ≤ 20) = 1 - e^(-20/25)
= 1 - e^(-4/5)
≈ 0.3297
Hence, the probability that a carryout order will be ready within 20 minutes is approximately 0.3297.
2. Probability that the order will not be ready after 30 minutes:
To calculate this probability, we can use the complementary cumulative distribution function (CCDF) or 1 - CDF.
P(X > 30) = 1 - P(X ≤ 30)
= 1 - (1 - e^(-30/25))
= e^(-6/5)
≈ 0.3012
Therefore, the probability that the order will not be ready after 30 minutes is approximately 0.3012.
3. Probability of picking up the order and returning home within 40 minutes:
To calculate this probability, we need to consider the time it takes for the customer to drive to the café, pick up the order, and return home.
The total time needed is the sum of the time it takes to pick up the order and the time it takes to drive home. Since the time it takes for the order to be ready follows an exponential distribution, the sum of two exponential random variables also follows an exponential distribution with a rate equal to the sum of the individual rates.
Let's define X as the time it takes to drive to the café and Y as the time it takes to pick up the order. We know that X has a mean of 15 minutes, and Y has a mean of 25 minutes.
The mean of the sum, X + Y, is the sum of the means, which is 15 + 25 = 40 minutes.
Therefore, the rate parameter λ for the sum is the reciprocal of the mean, λ = 1/40.
Now we can calculate the probability:
P(X + Y ≤ 40) = 1 - e^(-40/40)
= 1 - e^(-1)
≈ 0.6321
So, the probability that the customer can drive to the café, pick up the order, and return home by 6:00 PM is approximately 0.6321.