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 find the probability that a carryout order will be ready within 20 minutes, we can use the exponential distribution formula and plug in the given mean of 25 minutes.
1. P(X ≤ 20) = 1 - e^(-λx), where λ is the rate parameter.
Since the mean is 25 minutes, we can calculate the rate parameter using the formula λ = 1/mean.
λ = 1/25
Now, we can substitute the values in the formula:
P(X ≤ 20) = 1 - e^(-λx)
P(X ≤ 20) = 1 - e^(-(1/25)*20)
P(X ≤ 20) = 1 - e^(-0.8)
Calculating this expression:
P(X ≤ 20) ≈ 0.5488
Therefore, the probability that a carryout order will be ready within 20 minutes is approximately 0.5488 (to 4 decimals).
Next, to find the probability that the order will not be ready when a customer arrives 30 minutes after placing the order, we can use the exponential distribution formula again.
2. P(X > 30) = e^(-λx), where λ is the rate parameter.
Using the same rate parameter (λ = 1/25), we substitute the values:
P(X > 30) = e^(-λx)
P(X > 30) = e^(-(1/25)*30)
P(X > 30) = e^(-1.2)
Calculating this expression:
P(X > 30) ≈ 0.3012
Therefore, the probability that the order will not be ready when a customer arrives 30 minutes after placing the order is approximately 0.3012 (to 4 decimals).
Lastly, to find the probability that the customer can drive to the café, pick up the order, and return home by 6:00 P.M. (40 minutes), we need to calculate the cumulative distribution function (CDF).
3. P(X ≤ 40) = 1 - e^(-λx), where λ is the rate parameter.
Using the same rate parameter (λ = 1/25), we substitute the values:
P(X ≤ 40) = 1 - e^(-λx)
P(X ≤ 40) = 1 - e^(-(1/25)*40)
P(X ≤ 40) = 1 - e^(-1.6)
Calculating this expression:
P(X ≤ 40) ≈ 0.7026
Therefore, the probability that the customer can drive to the café, pick up the order, and return home by 6:00 P.M. is approximately 0.7026 (to 4 decimals).
To answer these questions, we can use the exponential distribution formula. The exponential distribution is commonly used to model the time between events occurring at a constant rate.
The exponential distribution formula is given by:
f(x; λ) = λ * e^(-λx)
Where:
- f(x; λ) is the probability density function (PDF) of the exponential distribution for a given value x
- λ is the rate parameter, which is equal to 1/mean in the case of the exponential distribution
- e is the base of the natural logarithm, approximately 2.71828
For the first question, we need to find the probability that a carryout order will be ready within 20 minutes. We are given that the mean is 25 minutes, so the rate parameter λ is equal to 1/25.
Plugging the values into the formula:
f(x; λ) = (1/25) * e^(-(1/25)*20)
Calculating this expression will give us the probability.
For the second question, we want to find the probability that an order will not be ready if a customer arrives 30 minutes after placing the order. The probability of an order not being ready is the complement of the probability that it will be ready within 30 minutes. So, we can use the same formula from the first question, but with x = 30, and subtract the result from 1.
1 - f(30; λ) = 1 - [(1/25) * e^(-(1/25)*30)]
For the third question, we need to find the probability that the customer can drive to the café, pick up the order, and return home within 40 minutes (20 minutes to get to the café, 20 minutes for the order to be ready). We can use the cumulative distribution function (CDF) of the exponential distribution, which represents the probability that a random variable takes on a value less than or equal to a given value.
The CDF of the exponential distribution is given by:
F(x; λ) = 1 - e^(-λx)
Using this formula, we can calculate:
F(40; λ) = 1 - e^(-(1/25)*40)
Calculating this expression will give us the probability.