(c) If 5 of these machines are being repaired, what is the probability that at least one of them will take longer than 5 hours to repair? Problem 2 (15 Points) A surgical procedure has an
80%
of being successful each time it is performed. Assume each surgery is independent. (a) What is the probability that the next three surgeries will be unsuccessful? (b) What is the probability that at least eight of the next ten procedures are successful? (c) What is the expected number of successful surgeries before a failure occurs? The gamma distribution is a continuous distribution, one of whose purpose is to extend the usefulness of the exponential distribution in modeling waiting times. The gamma distribution has two parameters, shape
(α)
and rate
(λ)
. If
X
follows a gamma distribution, then
μ X
​
= λ
α
​
σ X
2
​
= λ 2
α
​
A company is interested in using one of two possible batteries for their next project. The lifetime of battery
A
follows a Normal distribution with mean 200 and standard deviation 25 . The lifetime of battery
B
follows a gamma distribution with shape
α=10
and rate
λ=0.05
. Note: The function
rgamma(n
, shape, rate) will draw a random sample of size
n
from the gamma distribution. (a) The company wishes to estimate the probability that battery
A
will last longer than battery
B
. Use a simulated sample of size 10,000 to estimate the probability that battery
A
lasts longer than battery
B
. (b) The company will buy 10 batteries of each type. Estimate, using simulation, the probability that the average lifetime of the 10 batteries of type
A
will last longer than the average lifetime of the 10 batteries of type
B
. (c) Using the Central Limit Theorem, calculate the probability from part (b). Bonus Question (5 points) A company is manufacturing highway emergency flares. Such flares are supposed to burn for an average of 20 minutes. Every hour, a quality control test is conducted by collecting a sample of 5 flares from those produced and determining their average burn time. If the manufacturing process is working correctly, there is a
68%
chance that the average burn time of a flare is between 17 and 25 minutes. The quality engineer in charge of the process believes that if 4 or more of the 5 sampled flares fall outside these bounds, then this is a signal that the process might not be performing as expected. If the manufacturing process is working correctly, what is the probability that the quality engineer in charge will conclude the process might not be performing as expected during the next quality control test? The Rockwell hardness of certain metal pins is known to have a mean
μ=50
and a standard deviation
σ=4
. (a) Assume that the distribution of all such pin hardness measurements is known to be normal. If we randomly select 1 pin from the population, what is the probability that the hardness is less than 46 ? (b) Assume that the distribution of all such pin hardness measurements is known to be normal. If now a random sample of 16 pins is obtained, what is the probability that the average hardness is less than 46 ? (c) Assume that the distribution of all such pin hardness measurements is known to be normal. If now a random sample of 64 pins is obtained, what is the probability that the average hardness is less than 46 ? Problem 4 (10 Points) Two-dimensional Poisson process. The number of plants of a certain species in a certain forest has a Poisson distribution with mean 10 plants per acre. The number of plants in
T
acres therefore has a Poisson distribution with mean
10T
. (a) What is the probability that there will be exactly 18 plants in a two-acre region? (b) What is the probability that there will be exactly 12 plants in a circle with radius
100ft
? (1 acre
=43,560ft 2
)