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)?
Well, it looks like we have some probability problems involving Collina's Italian Café. Let's dive right in and add a sprinkle of humor along the way!
1. To find the probability that a carryout order will be ready within 20 minutes, we'll use the exponential distribution. Since the mean is 25 minutes, we can calculate the probability using the formula P(X ≤ x) = 1 - e^(-λx), where λ is the rate parameter. In this case, λ = 1/25.
P(X ≤ 20) = 1 - e^(-1/25 * 20)
≈ 1 - e^(-0.8)
≈ 0.5518
So, the probability that a carryout order will be ready within 20 minutes is approximately 0.5518.
2. Now, let's calculate the probability that the order will not be ready after 30 minutes. Since the exponential distribution is memoryless, we can calculate it as P(X > 30) = 1 - P(X ≤ 30).
P(X > 30) = 1 - (1 - e^(-1/25 * 30))
≈ 1 - (1 - e^(-1.2))
≈ 0.3012
Therefore, the probability that the order will not be ready after 30 minutes is approximately 0.3012.
3. Finally, let's determine the probability that our customer can make it to the café, pick up the order, and return home within 40 minutes (from 5:20 PM to 6:00 PM). We need to find P(X ≤ 40 - 15), since the customer lives 15 minutes away.
P(X ≤ 25) = 1 - e^(-1/25 * 25)
≈ 1 - e^(-1)
≈ 0.6321
Therefore, the probability that the customer can complete the trip within 40 minutes is approximately 0.6321.
Remember, these results are approximate! Now go grab your delicious order and enjoy the tasty goodness of Collina's Italian Café!
To answer these questions, we will use the exponential distribution formula. Let's define the parameters:
λ = 1/mean = 1/25 = 0.04
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 20 minutes.
P(X ≤ x) = 1 - e^(-λx)
Substituting the values:
P(X ≤ 20) = 1 - e^(-0.04 * 20)
Calculating this value, we find that the probability is approximately 0.3297.
2. Probability that the order will not be ready after 30 minutes:
To find this probability, we need to calculate the survival function or complementary cumulative distribution function (CCDF) at 30 minutes.
P(X > x) = e^(-λx)
Substituting the values:
P(X > 30) = e^(-0.04 * 30)
Calculating this value, we find that the probability is approximately 0.2231.
3. Probability that the customer can complete the entire process within 40 minutes:
To calculate this probability, we need to find the difference between the CDF at 40 minutes and the CDF at 15 minutes.
P(15 < X ≤ 40) = P(X ≤ 40) - P(X ≤ 15)
Using the exponential distribution formula, we get:
P(15 < X ≤ 40) = 1 - e^(-0.04 * 40) - (1 - e^(-0.04 * 15))
Calculating this value, we find that the probability is approximately 0.5850.
Please note that these values are rounded to four decimal places.