1. The analysis of shafts for a compressor is classiﬁed in terms of the machine tool that was used for manufacturing the shafts and by conformance to speciﬁcations, as shown below. Tool 1 surface ﬁnish conforms

yes no

roundness roundness conforms yes no 645 10 23 19

yes no

roundness roundness conforms yes no 574 18 12 3 55

Tool 2 surface ﬁnish conforms

(a) If a shaft is selected at random, what is the probability that the shaft conforms to surface ﬁnish requirements or to roundness requirements or is from Tool 1? (b) If a shaft is selected at random, what is the probability that the shaft conforms to surface ﬁnish requirements or does not conform to roundness requirements or is from Tool 2? (c) If a shaft is selected at random, what is the probability that the shaft conforms to both the surface ﬁnish and roundness requirements or the shaft is from Tool 1? (d) If a shaft is selected at random, what is the probability that the shaft conforms to surface ﬁnish requirements or the shaft is from Tool 2? (e) Given that a shaft selected at random conforms to surface ﬁnish requirements, what is the probability that it is from Tool 1? (f)) Given that a shaft selected at random conforms to surface ﬁnish requirements, what is the (f probability that it conforms to roundness requirements? (g) Given that a shaft selected at random does not conform to surface ﬁnish requirements, what is the probability that it is from Tool 1 and does not conform to roundness requirements? 2. A batch of 1000 containers for frozen orange juice contains contains 13 that are defective. defective. Three containers containers are selected at random, without replacement, from the batch. (a) What is the probability that the third one selected is defective given that the ﬁrst and the second one selected were defective? (b) What is the probability that the third one selected is defective given that the ﬁrst one selected was defective and the second one selected was okay? (c) What is the probability that all the three were defective? 3. A batch of 100 injection-molded parts contains 12 that have suﬀered excessive shrinkage. (a) If two parts are selected at random, and without replacement, what is the probability that the second part selected is one with excessive linkage? (b) If three parts are selected at random, and without replacement, what is the probability that the third part selected is one with excessive linkage? 1

2

PROBLEM SET 5 PROBABLITY

4. Each of the following circuits operates only if there is a path of functioning devices from left to right. The probability that each device functions is shown on the graph. Assume that devices fail independently. What is the probability that the each of the circuits operates? a

(b)

d b e

c (a) 0.9

0.8

0.7

f 0.95

h

m

0.95

i

g

Figure 1

5. Customers are used to evaluate preliminary product designs. In the past, 96% of highly successful products received good reviews, 66% of moderately successful products received good reviews, and 12% of poor products received good reviews. In addition, 45% of products have been highly successful, 37% have been moderately successful, and 18% have been poor products. (a) What is the probability that a product attains a good review? (b) If a new design attains a good review, what is the probability that it will be a highly successful product? (c) If a product does not attain a good review, what is the probability it will be a highly successful product? 6. An inspector working for a company has a 98% chance of correctly identifying defective items and a 1% chance of incorrectly classifying a good item as defective. The company has evidence that its line produces 2% of nonconforming items. (a) What is the probability that an item selected for inspection is classiﬁed as defective? (b) If an item selected at random is classiﬁed as non-defective, what is the probability that it is indeed good? 7. Let X be a random variable with probability distribution function given by f (0) =

1 1 1 1 1 , f (1) = , f (2) = , . . . f ( 9) = 10 , f (10) = 10 2 4 8 2 2

(a) Find the mean of X . (b) Find the variance and standard deviation of X . (c) What is P (3 X 8)?

≤ ≤

PROBLEM SET 5 PROBABLITY

3

8. Suppose that X is a discrete uniform random variable on the integers 11 through 20. Determine the mean, variance and the standard deviation of the random variable Y = 2X 5 and compare to those of X .

−

9. Let X denotes the number of bits received in error in a digital communication channel, and assume that X is a binomial random variable with p = 0.0014. If 1500 bits are transmitted, determine the following. (a) The probability that there is exactly one error received. (b) The probability that no error is received. (c) The probability that at least one error is received. (d) The probability that not more than 3 errors are received. (e) The mean and variance of X . 10. Because not all airline passengers show up for their reserved seat, an airline sells 325 tickets for a ﬂight that holds only 320 passengers. The probability that a passenger does not show up is 0.025, and the passengers behave independently. (a) What is the probability that every passenger who shows up can take the ﬂight? (b) What is the probability that the ﬂight departs with empty seats? 11. Suppose that the number of customers X that enter a cafe in twenty minutes is a Poisson random variable, and the probability that no customer enter the cafe in an hour is 0.02. (a) Find the probability distribution function of X . (b) Find the mean and variance of the number of customers entering the cafe in twenty minutes and in an hour. (c) What is the probability that no customer enter the cafe in twenty minutes? (d) What is the probability that there will be at least 3 customers entering the cafe in twenty minutes? 12. The number of ﬂaws in bolts of cloth in textile manufacturing is assumed to be Poisson distributed with a mean of 0.05 ﬂaw per square meter. (a) What is the probability that there are at least two ﬂaws in 1 square meter of cloth? (b) What is the probability that there are no ﬂaw in 5 square meters of cloth? (c) What is the probability that there are 3 ﬂaws in 15 square meters of cloth? (d) What is the largest possible area of cloth such that the probability that no ﬂaw is found on the area is larger than 0.5? 13. Suppose that X is a continuous random variable with probability density function

√

3 x 0 x 100 f (x) = , 2000 (a) Find the cumulative distribution function of X . (b) Find P (10 X 50). (c) Find the mean, variance and standard deviation of X .

≤ ≤

≤ ≤

14. Let Z be a standard normal random variable. Find the following probabilities. (a) P (Z < 1.32)

(b) P (Z > 1.32)

(c) P (Z <

(e) P (0 < Z < 1.32)

(f) P ( Z < 1.32)

(g) P (1 < Z < 3)

| |

−1.32)

(d) P (Z >

−1.32)

(h) P ( 1 < Z < 3)

−

4

PROBLEM SET 5 PROBABLITY

15. Let Z be a standard normal random variable. Determine the value of z that satisﬁes each of the following. (a) P (Z < z ) = 0.9

(b) P (Z > z ) = 0.4

(d) P ( Z < z ) = 0.8

(e) P ( Z > z ) = 0.05

| |

| |

(c) P (Z > z ) = 0.6 (f) P ( 1.32 < Z < z ) = 0.6

−

16. The compressive strength of samples of cement can be modeled by a normal distribution with a mean of 7000 kilometers per square centimeter and a standard deviation of 180 kilograms per square centimeter. (a) What is the probability that a sample’s strength is less than 7200 kilograms per centimeter? (b) What is the probability that a sample’s strength is between 6800 and 6900 kilograms per centimeter? (c) What is the probability that a sample’s strength is between 6950 and 7050 kilograms per centimeter? (d) What strength is exceeded by 98% of the samples? 17. The diameter of the dot produced by a printer is normally distributed with mean diameter 5mm and standard deviation of 0.2mm. (a) What is the probability that the diameter of a dot exceeds 5.5mm? (b) What is the probability that a diameter is between 4.9mm and 5.1mm? (c) What standard deviation of diameter is needed so that the probability in part (b) is 0.999? 18. The diameter of a shaft in an optical storage drive is normally distributed with mean µcm and standard deviation 0.001cm. The speciﬁcations on the shaft are 0.7 0.003cm. What proportion of shafts conforms to speciﬁcation if (a) µ = 0.7015 (b) µ = 0.7?

±

19. Suppose that X has an exponential distribution with λ = 1.2. Determine the following. (a) The mean µ and standard deviation σ of X . (b) P (X < 5) (c) P (X > 5) (d) P (µ < X < µ + σ) (e) P (µ σ < X < µ) (f) P (µ σ < X < µ + σ ) (g) The value x such that P (µ < X < x) = 0.2 (h) The value x such that P (x < X < µ) = 0.2

−

−

20. The time between calls to a plumbing supply business is exponentially distributed with a mean time between calls of 12 minutes. (a) What is the probability that there are no calls within a 30-minute interval? (b) What is the probability that at least one call arrives within a 10-minute interval? (c) What is the probability that the ﬁrst call arrives between 10 and 20 minutes after opening? (d) Determine the length of an interval of time such that the probability of at least one call in the interval is 0.95. 21. The time between arrival of electronic messages at your computer is exponentially distributed with a mean of 1.5 hours. (a) What is the probability that you do not receive a message during a 1.5-hour period? (b) What is the probability that you do not receive a message during a 4 hour period? (c) If you had not received a message during the last three hours, What is the probability that you do not receive a message in the next 3 hours? (d) What is the expected time between your second and third message?

PROBLEM SET 5 PROBABLITY

5

Answer:

1412 1444 1271 1425 655 (b) (c) (d) (e) 1467 1467 1467 1467 1247 11 12 13 12 11 2. (a) (b) (c) 998 998 1000 999 998 12 12 3. (a) (b) 100 100 4. (a) 0.9293 (b) (a + b + c ab ac bc + abc)(d + e de) + ( f + g f g )hi (a + b + c ab ac bc + abc)(d + e de)(f + g f g)hi 1. (a)

(f)

1219 1247

(g)

× × × ×

− − − − − −

−

5. (a) 0.6978

−

(b) 0.6190

6. (a) 0.0294

(b) 0.9996

7. (a) 0.9990

(b)1.9795

8. X Y

mean = 15.5, mean = 26,

9. (a) 0.2571

(c) 0.0596 (c) 0.1230

variance = 8.25 variance = 33

(b)0.1223

10. (a) 0.9101

−

− −

standard deviation = 2.8723 standard deviation = 5.7446

(c) 0.8777

(d) 0.8388

(e) 2.1, 2.0971

(b) 0.8234 n −λ λ

11. (a) f (n) = e

n!

, λ = 3.9120, n = 0, 1, 2, 3, . . .

(b) In twenty minutes, mean=variance= 1.3040, in an hour, mean=variance= 3.9120 (c) 0.2714 (d) 0.1438 12. (a) 0.0012

0 ( )= 1000 1

(b) 0.7788 x<0

,

13. (a) F x

3/2

x

,

0

15. (a) 1.28

4800 (c) 60, , 40 7

(b) 0.3219

≤ x ≤ 100

(b) 0.0934 (f) 0.8132

(b) 0.25

(c)

(c) 0.0934 (g) 0.1573

− 0.25

(d) 1.28

16. (a) 0.8665

(b) 0.1542

(c) 0.2205

17. (a) 0.0062

(b) 0.3829

(c) 0.0304 mm

18. (a) 0.9332

(b) 0.9973

19. (a) 0.8333 (e) 0.5654

(d) 13.8629

x > 100

,

14. (a) 0.9066 (e) 0.4066

(c) 0.0332

(b) 0.9975 (f) 0.8111

(d) 0.9066 (h) 0.8400 (e) 1.96

(f) 0.51

(d) 6631 kilograms per centimeter

(c) 0.0025 (g) 1.4499

(d) 0.2457 (h) 0.4548

20. (a) 0.0821

(b) 0.5654

(c) 0.2457

(d) 35.9488 minutes

21. (a) 0.3679

(b) 0.0695

(c) 0.1353

(d) 1.5 hours

3 7

19 220

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