Pt2520 Unit 1 Practice Midterm

STAT220

Spring 2012
Practice Midterm

1. Let A and B be events such that P(A) = 0.25 and P(B) = 0.16. Find P(A|B) if
(a) P(A B) = 0.06
(b) P(B|A) = 0.4
(c) P(AB) = 0.33
(d) A and B are independent
(e) A and B are mutually exclusive
2. Among employees of a certain firm, 70% know C/C++, 60% know Java, and 90% know at least one of the two languages. (a) What is the probability that a selected programmer knows both languages?
(b) What is the probability that a selected programmer knows C/C++ but not Java?
(c) What is the probability that a selected programmer knows only one of the two languages?
(d) If a programmer knows Java, what is the probability that he/she knows C/C++?
(e) If a programmer knows C/C++, what is the probability that he/she
knows Java?
(f) Are the events “know Java” and “know C/C++” independent? Are then mutually exclusive? Explain.
3. A maintenance firm has gathered the following information regarding the failure mechanisms for air conditioning systems
Evidence of gas leaks
Evidence of electrical failure Yes
No
Yes
55
17
No
32
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The units without evidence of gas leaks or electrical failure showed other types of failure. If this is a representative sample of AC failure, find the probability
(a) That failure involves a gas leak
(b) That there is evidence of electrical failure given that there was a gas leak
(c) That there is evidence of a gas leak given that there is evidence of electrical failure
4. The following circuit operates if and only if there is a path of functional devices from left to right. Assume devices fail independently and that the probability of failure of each device is as shown. What is the probability that the circuit operates?

5. At an electronics plant, it is known from past experience that the probability is 0.86 that a new worker who has attended the company's training program will meet the production quota and that the corresponding probability is 0.35 for a new worker who has not attended the company's training program. Assume that 80% of all new workers attend the training program.
(a) What is the probability that a new worker will meet the production quota?
(b) What is the probability that a worker who meets the production quota attended the company's training program? 6. A shipment of 7 television sets contains 2 defective sets. A hotel makes a random purchase of 3 of the sets. If
X is the number of defective sets purchased by the hotel, find the probability distribution of X.
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7. The following table contains the probability distribution for the number of traffic accidents daily in a small city. x
0
1
2
3
4
5 p(x) 0.10
0.20
0.45
0.15
0.05
0.05
a) Find the cumulative distribution function of X.
b) Compute the mean and the variance of X.
8. The number of requests for assistance received by a towing service is a Poisson process with rate of 4 per hour. (a) Compute the probability that exactly ten requests are received during a particular 2-hour period.
(b) If the operators of the towing service take a 30-min break for lunch, what is the probability that they do not miss any calls for assistance?
9. According to Chemical Engineering Progress (November 1990), approximately 30% of all pipework failures in chemical plants are caused by operator error.
(a) What is the probability that out of the next 20 pipework failures at least 10 are due to operator error?
(b) What is the probability that no more than 4 out of 20 such failures are due to operator error?
(c) Among the next 20 pipework failures, find the mean and the standard deviation of the number of failures caused by operator error.
(d) Suppose, for a particular plant, that out of the random sample of 20 such failures, exactly 5 are due to operator error. Do you feel that the 30% figure stated above applies to this plant? Comment.
10. The weekly repair cost (in $100), X, for a certain machine has a probability density function given by
cx(1  x) 0  x  1 f ( x)   otherwise  0
(a) Find the value of c that makes f(x) a valid probability density function.
(b) Find and sketch the cumulative distribution function of X.
(c) What is the probability that repair costs will exceed $75 during a week?
(d) Find the mean and variance of the repair costs.
11. The life expectancy X, in years, of a mechanical device has cumulative distribution function
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F ( x)  1  3 for x  3 x (a) Find the probability density function f(x) for X.
(b) Find the mean life expectancy of the device.
(c) Find the probability that the device lasts at least 6 years.
(d) Find the probability that the device lasts at least 6 years given that it has lasted 4 years.
12. Suppose the time it takes a data collection operator to fill out an electronic form for a database is uniformly between 1.5 and 2.2 minutes.
(a) What is the mean and variance of the time it takes an operator to fill out the form?
(b) What is the probability that it will take less than two minutes to fill out the form?
13. The compressive strength of samples of cement can be modeled by a normal distribution with a mean of 6000 kilograms per square centimeter (Kg/cm2) and a standard deviation of 100 Kg/cm2.
(a) What is the probability that a sample’s strength is less than 6250 Kg/cm2?
(b) What is the probability that a sample’s strength is between 5800 and 5900 Kg/cm2?
(c) What strength is exceeded by 95% of the samples?
(d) What is the probability that the mean strength of 16 samples exceeds 6050 Kg/cm2?
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14. In a grinding operation, there is an upper specification of 3.150 in. on a dimension of a certain part after grinding. Suppose that the standard deviation of this normally distributed dimension for parts of this type ground to any particular mean dimension μ is σ = 0.002 in. Suppose further that you desire to have no more than 3% of the parts fail to meet specifications. What is the maximum μ that can be used if this 3% requirement is to be met?
15.
Suppose the lifetime of a certain type of component is a random variable having an exponential distribution with mean lifetime 5 years.
(a) What is the probability that a selected component of this type will last more than 4 years?
(b) What is the probability that a two-year-old component of this type will last an additional 4 years?
(c) What is the 25th percentile of the lifetimes of such components?
(d) What is the probability that the mean lifetime of 64 randomly selected components is less than 4.8 years?
16. In the Ames Bank (open 24h every day) 5 customers arrive on average during an hour. For the following questions state each time which random variable you use and what distribution assumption you make.
(a) What is the probability that during an hour no customer arrives?
(b) What is the probability that during an hour more than 7 customers arrive?
(c) What is the probability that there’s more than 30 minutes between the 2nd and 3rd customer on New
Year’s Day?
(d) Starting at some time 0. What is the probability that the first customer arrives after exactly 10 min?
Within the first ten minutes?
(e) How many minutes do you expect to wait on average between arrivals?
(f) How many customers do you expect to arrive within 3 hours?
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