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)?
Jim Tree pays for shipping on the basis of weight. He knows weight is normally distributed so he can use the standard normal distribution. He decides to weigh 1,000 randomly trees in his shipment. Next, he calculates the mean and standard deviation of their weights. He finds the mean is 30 lbs and the standard deviation is 10 lbs. Now, Jim uses the normal distribution table to calculate the number of trees in each segment of the distribution.
To find the probability that a carryout order will be ready within 20 minutes, we need to use the exponential distribution formula.
The formula for the exponential distribution is:
P(x < t) = 1 - e^(-λt)
Where:
P(x < t) is the probability that the value of the random variable is less than t,
e is the base of the natural logarithm (approximately 2.71828),
λ is the rate parameter (in this case, λ = 1/25 because the mean is 25 minutes), and
t is the specific value we're interested in (in this case, t = 20 minutes).
Applying the formula, we get:
P(x < 20) = 1 - e^(-1/25 * 20)
Calculating the value:
P(x < 20) = 1 - e^(-0.8)
Using a calculator or software, we find:
P(x < 20) ≈ 0.5507
Therefore, the probability that a carryout order will be ready within 20 minutes is approximately 0.5507 (to 4 decimals).
To find the probability that a carryout order will be ready within 20 minutes, we can use the exponential distribution formula.
The exponential distribution probability density function is given by:
f(x) = λ * e^(-λx)
where λ is the rate parameter (equal to 1/mean) and x is the time value.
In this case, the mean is 25 minutes, so λ = 1/25.
To calculate the probability, we need to find the cumulative distribution function (CDF), which represents the probability that the order will be ready in less than or equal to a given time.
The CDF for the exponential distribution is given by:
F(x) = 1 - e^(-λx)
To find the probability of the order being ready within 20 minutes, we substitute x = 20 into the CDF formula:
F(20) = 1 - e^(-λ * 20)
Substituting λ = 1/25, we get:
F(20) = 1 - e^(-1/25 * 20)
Now, we can calculate this probability using a calculator or spreadsheet.
F(20) ≈ 0.4561
So, the probability that a carryout order will be ready within 20 minutes is approximately 0.4561 (rounded to 4 decimals).
You might try this formula:
Pr[T ≤ t] = 1 - e^-ht
h = reciprocal of the mean
t = time
Therefore, the reciprocal of 25 is 1/25 or .04
Substituting into the formula:
Pr[T ≤ 20] = 1 - e^-.04(20) = 1 - e^-.8 = 1 - .4493 = .5507
Check these calculations.