1. Chapter 5: Discrete Distributions 1
Chapter 5
Discrete Distributions
LEARNING OBJECTIVES
The overall learning objective of Chapter 5 is to help you understand a category of probability
distributions that produces only discrete outcomes, thereby enabling you to:
1. Distinguish between discrete random variables and continuous random variables.
2. Know how to determine the mean and variance of a discrete distribution.
3. Identify the type of statistical experiments that can be described by the binomial
distribution and know how to work such problems.
4. Decide when to use the Poisson distribution in analyzing statistical experiments and
know how to work such problems.
.
5. Decide when binomial distribution problems can be approximated by the Poisson
distribution and know how to work such problems.
6. Decide when to use the hypergeometric distribution and know how to work such
problems
CHAPTER TEACHING STRATEGY
Chapters 5 and 6 introduce the student to several statistical distributions. It is important
to differentiate between the discrete distributions of chapter 5 and the continuous distributions of
chapter 6.
The approach taken in presenting the binomial distribution is to build on techniques
presented in chapter 4. It can be helpful to take the time to apply the law of multiplication for
independent events to a problem and demonstrate to students that sequence is important. From
there, the student will more easily understand that by using combinations, one can more quickly
determine the number of sequences and weigh the probability of obtaining a single sequence by
that number. In a sense, we are developing the binomial formula through an inductive process.
2. Chapter 5: Discrete Distributions 2
Thus, the binomial formula becomes more of a summary device than a statistical "trick". The
binomial tables presented in this text are noncumulative. This makes it easier for the student to
recognize that the table is but a listing of a series of binomial formula computations. In addition,
it lends itself more readily to the graphing of a binomial distribution.
It is important to differentiate applications of the Poisson distribution from binomial
distribution problems. It is often difficult for students to determine which type of distribution to
apply to a problem. The Poisson distribution applies to rare occurrences over some interval. The
parameters involved in the binomial distribution (n and p) are different from the parameter
(Lambda) of a Poisson distribution.
It is sometimes difficult for students to know how to handle Poisson problems where the
interval for the problem is different than the interval for which Lambda was developed. If they
can view Lambda as a long run average which can be appropriately adjusted for various
intervals, then students can be more successful with these types of problems.
Solving for the mean and standard deviation of binomial distributions prepares the
students for chapter 6 where the normal distribution is sometimes used to solve binomial
distribution problems. In addition, graphing binomial and Poisson distributions affords the
student the opportunity to visualize the meaning and impact of a particular set of parameters for a
distribution.
It can be emphasized that the hypergeometric distribution is an exact distribution.
However, it is cumbersome to determine probabilities using the hypergeometric formula
particularly when computing cumulative probabilities. The hypergeometric distribution can be
presented as a fall-back position to be used when the binomial distribution should not be applied
because of the non independence of trials and size of sample.
CHAPTER OUTLINE
5.1 Discrete Versus Continuous Distributions
5.2 Describing a Discrete Distribution
Mean, Variance, and Standard Deviation of Discrete Distributions
Mean or Expected Value
Variance and Standard Deviation of a Discrete Distribution
5.3 Binomial Distribution
Solving a Binomial Problem
Using the Binomial Table
Using the Computer to Produce a Binomial Distribution
Mean and Standard Deviation of the Binomial Distribution
Graphing Binomial Distributions
3. Chapter 5: Discrete Distributions 3
5.4 Poisson Distribution
Working Poisson Problems by Formula
Using the Poisson Tables
Mean and Standard Deviation of a Poisson Distribution
Graphing Poisson Distributions
Using the Computer to Generate Poisson Distributions
Approximating Binomial Problems by the Poisson Distribution
5.5 Hypergeometric Distribution
Using the Computer to Solve for Hypergeometric Distribution
Probabilities
KEY TERMS
Binomial Distribution Hypergeometric Distribution
Continuous Distributions Lambda (λ)
Continuous Random Variables Mean, or Expected Value
Discrete Distributions Poisson Distribution
Discrete Random Variables Random Variable
SOLUTIONS TO PROBLEMS IN CHAPTER 5
5.1 x P(x) x·P(x) (x-µ)2
(x-µ)2
·P(x)
1 .238 .238 2.775556 0.6605823
2 .290 .580 0.443556 0.1286312
3 .177 .531 0.111556 0.0197454
4 .158 .632 1.779556 0.2811700
5 .137 .685 5.447556 0.7463152
µ = [x·P(x)] = 2.666 σ2
= (x-µ)2
·P(x) = 1.836444
σ = 836444.1 = 1.355155
9. Chapter 5: Discrete Distributions 9
5.9
a) n = 20 p = .78 x = 14
20C14 (.78)14
(.22)6
= 38,760(.030855)(.00011338) = .1356
b) n = 20 p = .75 x = 20
20C20 (.75)20
(.25)0
= (1)(.0031712)(1) = .0032
c) n = 20 p = .70 x < 12
Use table A.2:
P(x=0) + P(x=1) + . . . + P(x=11)=
.000 + .000 + .000 + .000 + .000 + .000 + .000 +
.001 + .004 + .012 + .031 + .065 = .113
5.10 n = 16 p = .40
P(x > 9): from Table A.2:
x Prob
9 .084
10 .039
11 .014
12 .004
13 .001
.142
10. Chapter 5: Discrete Distributions 10
P(3 < x < 6):
x Prob
3 .047
4 .101
5 .162
6 .198
.508
n = 13 p = .88
P(x = 10) = 13C10(.88)10
(.12)3
= 286(.278500976)(.001728) = .1376
P(x = 13) = 13C13(.88)13
(.12)0
= (1)(.1897906171)(1) = .1898
Expected Value = µ = n p = 13(.88) = 11.44
5.11 n = 25 P = .60
a) x > 15
P(x > 15) = P(x = 15) + P(x = 16) + · · · + P(x = 25)
Using Table A.2 n = 25, p = .80
x Prob
15 .161
16 .151
17 .120
18 .080
19 .044
20 .020
21 .007
22 .002
.585
b) x > 20
from a): P(x > 20) = P(x = 21) + P(x = 22) + P(x = 23) +
P(x = 24) + P(x = 25) =
.007 + .002 + .000 + .000 + .000 = .009
11. Chapter 5: Discrete Distributions 11
c) P(x < 10)
from Table A.2, x Prob.
9 .009
8 .003
7 .001
<6 .000
.013
5.12
The highest probability values are for x = 15, 16, 14, 17, 13, 18, and 12.
The expected value is 25(.60) = 15. The standard deviation is 2.45.
15 + 2(2.45) = 15 + 4.90 gives a range that goes from 10.10 to 19.90. From
table A.2, the sum of the probabilities of the values in this range (11 through
19) is .936 or 93.6% of the values which compares quite favorably with the
95% suggested by the empirical rule.
5.13
n = 15 p = .20
a) P(x = 5) = 15C5(.20)5
(.80)10
=
3003(.00032)(.1073742) = .1032
b) P(x > 9): Using Table A.2
P(x = 10) + P(x = 11) + . . . + P(x = 15) =
12. Chapter 5: Discrete Distributions 12
.000 + .000 + . . . + .000 = .000
c) P(x = 0) = 15C0(.20)0
(.80)15
=
(1)(1)(.035184) = .0352
d) P(4 < x < 7): Using Table A.2
P(x = 4) + P(x = 5) + P(x = 6) + P(x = 7) =
.188 + .103 + .043 + .014 = .348
e)
5.14 n = 18
a) p =.30 µ = 18(.30) = 5.4
p = .34 µ = 18(.34) = 6.12
b) P(x > 8) n = 18 p = .30
from Table A.2
x Prob
8 .081
9 .039
10 .015
11 .005
12 .001
.141
13. Chapter 5: Discrete Distributions 13
c) n = 18 p = .34
P(2 < x < 4) = P(x = 2) + P(x = 3) + P(x = 4) =
18C2(.34)2
(.66)16
+ 18C3(.34)3
(.66)15
+ 18C4(.34)4
(.66)14
=
.0229 + .0630 + .1217 = .2076
d) n = 18 p = .30 x = 0
18C0(.30)0
(.70)18
= .00163
n = 18 p = .34 x = 0
18C0(.34)0
(.66)18
= .00056
The probability that none are in the $500,000 to $1,000,000 is higher because
there is a smaller percentage in that category which is closer to zero.
5.15 a) Prob(x=5λ = 2.3)=
(2.35
)(e-2.3
) = (64.36343)(.1002588) = .0538
5! (120)
b) Prob(x=2λ = 3.9) =
(3.92
)(e-3.9
) = (15.21)(.02024) = .1539
2! (2)
c) Prob(x < 3λ = 4.1) =
Prob(x=3) + Prob(x=2) + Prob(x=1) + Prob(x=0) =
(4.13
)(e-4.1
) = (68.921)(.016574) = .1904
3! 6
+ (4.12
)(e-4.1
) = (16.81)(.016573) = .1393
2! 2
+ (4.11
)(e-4.1
) = (4.1)(.016573) = .0679
1! 1
18. Chapter 5: Discrete Distributions 18
d) λ = 0.6 mean = 0.6 standard deviation = 6.0 = .775
x Prob
0 .5488
1 .3293
2 .0988
3 .0198
4 .0030
5 .0004
6 .0000
5.18 λ = 2.84 minutes
a) Prob(x=6 λ = 2.8)
from Table A.3 .0407
19. Chapter 5: Discrete Distributions 19
b) Prob(x=0 λ = 2.8) =
from Table A.3 .0608
c) Unable to meet demand if x > 44 minutes:
x Prob.
5 .0872
6 .0407
7 .0163
8 .0057
9 .0018
10 .0005
11 .0001
x > 4 .1523
.1523 probability of being unable to meet the demand.
Probability of meeting the demand = 1 - (.1523) = .8477
15.23% of the time a second window will need to be opened.
d) λ = 2.8 arrivals4 minutes
Prob(x=3) arrivals2 minutes = ??
Lambda must be changed to the same interval (½ the size)
New lambda=1.4 arrivals2 minutes
Prob(x=3) λ=1.4) = from Table A.3 = .1128
Prob(x > 5 8 minutes) = ??
Lambda must be changed to the same interval(twice the size):
New lambda= 5.6 arrivals8 minutes
Prob(x > 5 λ = 5.6):
21. Chapter 5: Discrete Distributions 21
e) P(x = 8 15 minutes)
λ = 10.5 15 minutes
P(x = 8 15 minutes) =
!8
))(5.10(
!
5.108 −−
=
⋅ e
X
eX λ
λ
= .1009
5.20 λ = 5.6 days3 weeks
a) Prob(x=0 λ = 5.6):
from Table A.3 = .0037
b) Prob(x=6 λ = 5.6):
from Table A.3 = .1584
c) Prob(x > 15 λ = 5.6):
x Prob.
15 .0005
16 .0002
17 .0001
x > 15 .0008
Because this probability is so low, if it actually occurred, the researcher would
actually have to question the Lambda value as too low for this period.
5.21 λ = 0.6 trips 1 year
a) Prob(x=0 λ = 0.6):
from Table A.3 = .5488
b) Prob(x=1 λ = 0.6):
from Table A.3 = .3293
c) Prob(x > 2 λ = 0.6):
22. Chapter 5: Discrete Distributions 22
from Table A.3 x Prob.
2 .0988
3 .0198
4 .0030
5 .0004
6 .0000
x > 2 .1220
d) Prob(x < 3 3 year period):
The interval length has been increased (3 times)
New Lambda = λ = 1.8 trips3 years
Prob(x < 3 λ = 1.8):
from Table A.3 x Prob.
0 .1653
1 .2975
2 .2678
3 .1607
x < 3 .8913
e) Prob(x=4 6 years):
The interval has been increased (6 times)
New Lambda = λ = 3.6 trips6 years
Prob(x=4 λ = 3.6):
from Table A.3 = .1912
5.22 λ = 1.2 collisions4 months
a) Prob(x=0 λ = 1.2):
from Table A.3 = .3012
b) Prob(x=2 2 months):
23. Chapter 5: Discrete Distributions 23
The interval has been decreased (by ½)
New Lambda = λ = 0.6 collisions2 months
Prob(x=2 λ = 0.6):
from Table A.3 = .0988
c) Prob (x < 1 collision6 months):
The interval length has been increased (by 1.5)
New Lambda = λ = 1.8 collisions6 months
Prob(x < 1 λ = 1.8):
from Table A.3 x Prob.
0 .1653
1 .2975
x < 1 .4628
The result is likely to happen almost half the time (46.26%). Ship channel and
weather conditions are about normal for this period. Safety awareness is
about normal for this period. There is no compelling reason to reject the
lambda value of 0.6 collisions per 4 months based on an outcome of 0 or 1
collisions per 6 months.
5.23 λ = 1.2 penscarton
a) Prob(x=0 λ = 1.2):
from Table A.3 = .3012
b) Prob(x > 8 λ = 1.2):
from Table A.3 = .0000
c) Prob(x > 3 λ = 1.2):
from Table A.3 x Prob.
4 .0260
5 .0062
24. Chapter 5: Discrete Distributions 24
6 .0012
7 .0002
8 .0000
x > 3 .0336
5.24 n = 100,000 p = .00004
Prob(x > 7 n = 100,000 p = .00004):
λ = µ = n⋅p = 100,000(.00004) = 4.0
Since n > 20 and n⋅p < 7, the Poisson approximation to this binomial problem is
close enough.
Prob(x > 7 λ = 4):
Using Table A.3 x Prob.
7 .0595
8 .0298
9 .0132
10 .0053
11 .0019
12 .0006
13 .0002
14 .0001
x > 7 .1106
Prob(x>10 λ = 4):
Using Table A.3 x Prob.
11 .0019
12 .0006
13 .0002
14 .0001
x > 10 .0028
Since getting more than 10 is a rare occurrence, this particular geographic region
appears to have a higher average rate than other regions. An investigation of
particular characteristics of this region might be warranted.
25. Chapter 5: Discrete Distributions 25
5.25 p = .009 n = 200
Use the Poisson Distribution:
λ = n⋅p = 200(.009) = 1.8
P(x > 6) from Table A.3 =
P(x = 6) + P(x = 7) + P(x = 8) + P(x = 9) + . . . =
.0078 + .0020 + .0005 + .0001 = .0104
P(x > 10) = .0000
P(x = 0) = .1653
P(x < 5) = P(x = 0) + P(x = 1) + P(x = 2) + P( x = 3) + P(x = 4) =
.1653 + .2975 + .2678 + .1607 + .0723 = .9636
5.26 n = 300, p = .01, λ = n(p) = 300(.01) = 3
a) Prob(x = 5):
Using λ = 3 and Table A.3 = .1008
b) Prob (x < 4) = Prob.(x = 0) + Prob.(x = 1) + Prob.(x = 2) + Prob.(x = 3) =
.0498 + .1494 + .2240 + .2240 = .6472
c) The expected number = µ = λ = 3
5.27 a) Prob(x = 3 N = 11, A = 8, n = 4)
330
)3)(56(
411
1338
=
⋅
C
CC
= .5091
b) Prob(x < 2)N = 15, A = 5, n = 6)
Prob(x = 1) + Prob (x = 0) =
615
51015
C
CC ⋅
+
615
61005
C
CC ⋅
=
5005
)210)(1(
5005
)252)(5(
+
26. Chapter 5: Discrete Distributions 26
.2517 + .0420 = .2937
c) Prob(x=0 N = 9, A = 2, n = 3)
84
)35)(1(
39
3702
=
⋅
C
CC
= .4167
d) Prob(x > 4 N = 20, A = 5, n = 7) =
Prob(x = 5) + Prob(x = 6) + Prob(x = 7) =
720
21555
C
CC ⋅
+
720
11565
C
CC ⋅
+
720
01575
C
CC ⋅
=
77520
)105)(1(
+ 5C6 (impossible) + 5C7(impossible) = .0014
5.28 N = 19 n = 6
a) P(x = 1 private) A = 11
132,27
)56)(11(
619
58111
=
⋅
C
CC
= .0227
b) P(x = 4 private)
132,27
)28)(330(
619
28411
=
⋅
C
CC
= .3406
c) P(x = 6 private)
132,27
)1)(462(
619
08611
=
⋅
C
CC
= .0170
d) P(x = 0 private)
132,27
)28)(1(
619
68011
=
⋅
C
CC
= .0010
27. Chapter 5: Discrete Distributions 27
5.29 N = 17 A = 8 n = 4
a) P(x = 0) =
417
4908
C
CC ⋅
=
2380
)126)(1(
= .0529
b) P(x = 4) =
417
0948
C
CC ⋅
=
2380
)1)(70(
= .0294
c) P(x = 2 non computer) =
417
2829
C
CC ⋅
=
2380
)28)(36(
= .4235
5.30 N = 20 A = 16 white N - A = 4 red n = 5
a) Prob(x = 4 white) =
520
14416
C
CC ⋅
=
15504
)4)(1820(
= .4696
b) Prob(x = 4 red) =
520
11644
C
CC ⋅
=
15504
)16)(1(
= .0010
c) Prob(x = 5 red) =
520
01654
C
CC ⋅
= .0000 because 4C5 is impossible to determine
The participant cannot draw 5 red beads if there are only 4 to draw from.
5.31 N = 10 n = 4
a) A = 3 x = 2
210
)21)(3(
410
2723
=
⋅
C
CC
= .30
b) A = 5 x = 0
210
)5)(1(
410
4505
=
⋅
C
CC
= .0238
c) A = 5 x = 3
210
)5)(10(
410
1535
=
⋅
C
CC
= .2381
28. Chapter 5: Discrete Distributions 28
5.32 N = 16 A = 4 defective n = 3
a) Prob(x = 0) =
560
)220)(1(
316
31204
=
⋅
C
CC
= .3929
b) Prob(x = 3) =
560
)1)(4(
316
01234
=
⋅
C
CC
= .0071
c) Prob(x > 2) = Prob(x=2) + Prob(x=3) =
316
11224
C
CC ⋅
+ .0071 (from part b.) =
560
)12)(6( + .0071 = .1286 + .0071 = .1357
d) Prob(x < 1) = Prob(x=1) + Prob(x=0) =
316
21214
C
CC ⋅
+ .3929 (from part a.) =
560
)66)(4(
+ .3929 = .4714 + .3929 = .8643
5.33 N = 18 A = 11 Hispanic n = 5
Prob(x < 1) = Prob(1) + Prob(0) =
518
47111
C
CC ⋅
+
518
57011
C
CC ⋅
=
8568
)21)(1(
8568
)35)(11(
+ = .0449 + .0025 = .0474
It is fairly unlikely that these results occur by chance. A researcher might
investigate further the causes of this result. Were officers selected based on
leadership, years of service, dedication, prejudice, or what?
5.34 a) Prob(x=4 n = 11 and p = .23)
11C4(.23)4
(.77)7
= 330(.0028)(.1605) = .1482
b) Prob(x > 1n = 6 and p = .50) =
1 - Prob(x < 1) = 1 - Prob(x = 0) =
1-6C0(.50)0
(.50)6
= 1-(1)(1)(.0156) = .9844
31. Chapter 5: Discrete Distributions 31
5.37 a) Prob(x = 3 λ = 1.8) = .1607
b) Prob(x < 5 λ = 3.3) =
Prob(x = 4) + Prob(x = 3) + Prob(x = 2) + Prob(x = 1) + Prob(x = 0) =
x Prob.
0 .0369
1 .1217
2 .2008
3 .2209
4 .1823
x < 5 .7626
c) Prob(x > 3 λ = 2.1) =
x Prob.
3 .1890
4 .0992
5 .0417
6 .0146
7 .0044
8 .0011
9 .0003
10 .0001
11 .0000
x > 5 .3504
d) Prob(2 < x < 5 λ = 4.2) = Prob(x=3) + Prob(x=4) + Prob(x=5) =
x Prob.
3 .1852
4 .1944
5 .1633
2 < x < 5 .5429
5.38 a) Prob(x = 3 N = 6, n = 4, A = 5) =
15
)1)(10(
46
1135
=
⋅
C
CC
= .6667
32. Chapter 5: Discrete Distributions 32
b) Prob(x < 1 N = 10, n = 3, A = 5) = Prob(x = 1) + Prob(x = 0) =
310
2515
C
CC ⋅
+
310
3505
C
CC ⋅
=
120
)10)(1(
120
)10)(5(
+
= .4167 + .0833 = .5000
c) Prob(x > 2 N = 13, n = 5, A = 3) =
Prob(x=2) + Prob(x=3) Note: only 3 x's in population
513
31023
C
CC ⋅
+
513
21033
C
CC ⋅
=
1287
)45)(1(
1287
)120)(3(
+
= .2797 + .0350 = .3147
5.39 n = 25 p = .20 retired
from Table A.2: P(x = 7) = .111
P(x > 10): P(x = 10) + P(x = 11) + . . . + P(x = 25) =
.012 + .004 + .001 = .017
Expected Value = µ = n⋅p = 25(.20) = 5
n = 20 p = .40 mutual funds
P(x = 8) = .180
P(x < 6) = P(x = 0) + P(x = 1) + . . . + P(x = 5) =
.000 + .000 + .003 +.012 + .035 + .075 = .125
P(x = 0) = .000
P(x > 12) = P(x = 12) + P(x = 13) + . . . + P(x = 20) =
.035 + .015 + .005 + .001 = .056
x = 8
Expected Number = µ = n p = 20(.40) = 8
33. Chapter 5: Discrete Distributions 33
5.40 λ = 3.2 cars2 hours
a) Prob(x=3) cars per 1 hour) = ??
The interval has been decreased by ½.
The new λ = 1.6 cars1 hour.
Prob(x = 3 λ = 1.6) = (from Table A.3) .1378
b) Prob(x = 0 cars per ½ hour) = ??
The interval has been decreased by ¼ the original amount.
The new λ = 0.8 cars½ hour.
Prob(x = 0 λ = 0.8) = (from Table A.3) .4493
c) Prob(x > 5 λ = 1.6) = (from Table A.3)
x Prob.
5 .0176
6 .0047
7 .0011
8 .0002
.0236
Either a rare event occurred or perhaps the long-run average, λ, has changed
(increased).
5.41 N = 32 A = 10 n = 12
a) P(x = 3) =
1232
922310
C
CC ⋅
=
840,792,225
)420,497)(120(
= .2644
b) P(x = 6) =
1232
622610
C
CC ⋅
=
840,792,225
)613,74)(210(
= .0694
c) P(x = 0) =
1232
1222010
C
CC ⋅
=
840,792,225
)646,646)(1(
= .0029
34. Chapter 5: Discrete Distributions 34
d) A = 22
P(7 < x < 9) =
1232
510722
C
CC ⋅
+
1232
410822
C
CC ⋅
+
1232
310922
C
CC ⋅
=
840,792,225
)120)(420,497(
840,792,225
)210)(770,319(
840,792,225
)252)(544,170(
++
= .1903 + .2974 + .2644 = .7521
5.42 λ = 1.4 defects 1 lot If x > 3, buyer rejects If x < 3, buyer accepts
Prob(x < 3 λ = 1.4) = (from Table A.3)
x Prob.
0 .2466
1 .3452
2 .2417
3 .1128
x < 3 .9463
5.43 a) n = 20 and p = .25
The expected number = µ = n⋅p = (20)(.25) = 5.00
b) Prob(x < 1 n = 20 and p = .25) =
Prob(x = 1) + Prob(x = 0) = 20C1(.25)1
(.75)19
+ 20C0(.25)0
(.75)20
= (20)(.25)(.00423) + (1)(1)(.0032) = .0212 +. 0032 = .0244
Since the probability is so low, the population of your state may have a lower
percentage of chronic heart conditions than those of other states.
5.44 a) Prob(x > 7 n = 10 and p = .70) = (from Table A.2):
x Prob.
8 .233
9 .121
10 .028
x > 7 .382
35. Chapter 5: Discrete Distributions 35
Expected number = µ = n⋅p = 10(.70) = 7
b) Expected number = µ = n⋅p = 15 (1/3) = 5
Prob(x=0⋅ n = 15 and p = 1/3) =
15C0( )0
( )15
= .0023
c) Prob(x = 7 n = 7 and p = .53) = 7C7(.53)7
(.47)0
= .0117
Probably the 53% figure is too low for this population since the probability of
this occurrence is so low (.0117).
5.45 n = 12
a.) Prob(x = 0 long hours):
p = .20 12C0(.20)0
(.80)12
= .0687
b.) Prob(x > 6) long hours):
p = .20
Using Table A.2: .016 + .003 + .001 = .020
c) Prob(x = 5 good financing):
p = .25, 12C5(.25)5
(.75)7
= .1032
d.) p = .19 (good plan), expected number = µ = n(p) = 12(.19) = 2.28
5.46 n = 100,000 p = .000014
Worked as a Poisson: λ = n⋅p = 100,000(.000014) = 1.4
a) P(x = 5):
from Table A.3 = .0111
b) P(x = 0):
36. Chapter 5: Discrete Distributions 36
from Table A.3 = .2466
c) P(x > 6): (from Table A.3)
x Prob
7 .0005
8 .0001
.0006
5.47 Prob(x < 3) n = 8 and p = .60) = (from Table A.2)
x Prob.
0 .001
1 .008
2 .041
3 .124
x < 3 .174
17.4% of the time in a sample of eight, three or fewer customers are walk-ins by
chance. Other reasons for such a low number of walk-ins might be that she is
retaining more old customers than before or perhaps a new competitor is
attracting walk-ins away from her.
5.48 n = 25 p = .20
a) Prob(x = 8 n = 25 and p = .20) = (from Table A.2) .062
b) Prob(x > 10) n=25 and p = .20) = (from Table A.2)
x Prob.
11 .004
12 .001
13 .000
x > 10 .005
c) Since such a result would only occur 0.5% of the time by chance, it is likely
that the analyst's list was not representative of the entire state of Idaho or the
20% figure for the Idaho census is not correct.
5.49 λ = 0.6 flats2000 miles
Prob(x = 0 λ = 0.6) = (from Table A.3) .5488
37. Chapter 5: Discrete Distributions 37
Prob(x > 3 λ = 0.6) = (from Table A.3)
x Prob.
3 .0198
4 .0030
5 .0004
x > 3 .0232
One trip is independent of the other.
Let F = flat tire and NF = no flat tire
P(NF1 _ NF2) = P(NF1) ⋅ P(NF2)
P(NF) = .5488
P(NF1 _ NF2) = (.5488)(.5488) = .3012
5.50 N = 25 n = 8
a) P(x = 1 in NY) A = 4
825
72114
C
CC ⋅
=
575,081,1
)280,116)(4(
= .4300
b) P(x = 4 in top 10) A = 10
575,081,1
)1365(210(
825
415410
=
⋅
C
CC
= .2650
c) P(x = 0 in California) A = 4
575,081,1
)490,203)(1(
825
82104
=
⋅
C
CC
= .1881
d) P(x = 3 with M) A = 3
575,081,1
)334,26)(1(
825
52233
=
⋅
C
CC
= .0243
5.51 N = 24 n = 6 A = 8
38. Chapter 5: Discrete Distributions 38
a) P(x = 6) =
596,134
)1)(28(
624
01668
=
⋅
C
CC
= .0002
b) P(x = 0) =
596,134
)8008)(1(
624
61608
=
⋅
C
CC
= .0595
d) A = 16 East Side
P(x = 3) =
596,134
)56)(560(
624
38316
=
⋅
C
CC
= .2330
5.52 n = 25 p = .20
Expected Value = µ = n⋅p = 25(.20) = 5
µ = 25(.20) = 5
σ = )80)(.20(.25=⋅⋅ qpn = 2
P(x > 12) = (from Table A.2)
x Prob
13 .0000
The values for x > 12 are so far away from the expected value that they are very
unlikely to occur.
P(x = 14) = 25C14(.20)14
(.80)11
= .000063 which is very unlikely.
If this value (x = 14) actually occurred, one would doubt the validity of the
p = .20 figure or one would have experienced a very rare event.
39. Chapter 5: Discrete Distributions 39
5.53 λ = 2.4 calls1 minute
a) Prob(x = 0 λ = 2.4) = (from Table A.3) .0907
b) Can handle x < 5 calls Cannot handle x > 5 calls
Prob(x > 5 λ = 2.4) = (from Table A.3)
x Prob.
6 .0241
7 .0083
8 .0025
9 .0007
10 .0002
11 .0000
x > 5 .0358
c) Prob(x = 3 calls 2 minutes)
The interval has been increased 2 times.
New Lambda = λ = 4.8 calls2 minutes.
from Table A.3: .1517
d) Prob(x < 1 calls15 seconds):
The interval has been decreased by ¼.
New Lambda = λ = 0.6 calls15 seconds.
Prob(x < 1 λ = 0.6) = (from Table A.3)
Prob(x = 1) = .3293
Prob(x = 0) = .5488
Prob(x < 1) = .8781
40. Chapter 5: Discrete Distributions 40
5.54 n = 160 p = .01
Working this problem as a Poisson problem:
a) Expected number = µ = n(p) = 160(.01) = 1.6
b) P(x > 8):
Using Table A.3: x Prob.
8 .0002
9 .0000
.0002
c) P(2 < x < 6):
Using Table A.3: x Prob.
2 .2584
3 .1378
4 .0551
5 .0176
6 .0047
P(2 < x < 6) .4736
5.55 p = .005 n = 1,000
λ = n⋅p = (1,000)(.005) = 5
a) P(x < 4) = P(x = 0) + P(x = 1) + P(x = 2) + P(x = 3) =
.0067 + .0337 + .0842 + .1404 = .265
b) P(x > 10) = P(x = 11) + P(x = 12) + . . . =
.0082 + .0034 + .0013 + .0005 + .0002 = .0136
c) P(x = 0) = .0067
41. Chapter 5: Discrete Distributions 41
5.56 n = 8 P = .36 x = 0 Women
8C0(.36)0
(.64)8
= (1)(1)(.0281475) = .0281
It is unlikely that a company would randomly hire 8 physicians from the U.S. pool
and none of them would be female. If this actually happened, the figures might
be used as evidence in a lawsuit.
5.57 N = 32
a) n = 5 x = 3 A = 10
376,201
)231)(120(
532
222310
=
⋅
C
CC
= .1377
b) n = 8 x < 2 A = 6
832
82606
C
CC ⋅
+
832
72616
C
CC ⋅
+
832
62626
C
CC ⋅
=
300,518,10
)760,38)(15(
300,518,10
)800,657)(6(
300,518,10
)275,562,1)(1(
++ = .1485 + .3752 + .0553 = .5790
c) n = 5 x = 2 p = 3/26 = .1154
5C2(.1154)2
(.8846)3
= (10)(.013317)(.692215) = .0922
5.58 N = 14 n = 4
a) P(x = 4 N = 14, n = 4, A = 10 Northside)
1001
)1((210(
414
04410
=
⋅
C
CC
= .2098
b) P(x = 4 N = 14, n = 4, A = 4 West)
1001
)1)(1(
414
01044
=
⋅
C
CC
= .0010
c) P(x = 2 N = 14, n = 4, A = 4 West)
42. Chapter 5: Discrete Distributions 42
1001
)45)(6(
414
21024
=
⋅
C
CC
= .2697
5.59 a) λ = 3.841,000
P(x = 0) =
!0
84.3 84.30 −
⋅e
= .0215
b) λ = 7.682,000
P(x = 6) =
720
)000461975)(.258.195,205(
!6
68.7 68.76
=
⋅ −
e
= .1317
c) λ = 1.61,000 and λ = 4.83,000
from Table A.3:
P(x < 7) = P(x = 0) + P(x = 1) + . . . + P(x = 6) =
.0082 + .0395 + .0948 + .1517 + .1820 + .1747 + .1398 = .7907
5.60
This is a binomial distribution with n = 15 and p = .36.
µ = n⋅p = 15(.36) = 5.4
σ = )64)(.36(.15 = 1.86
The most likely values are near the mean, 5.4. Note from the printout that the
most probable values are at x = 5 and x = 6 which are near the mean.
5.61 This printout contains the probabilities for various values of x from zero to eleven from a
Poisson distribution with λ = 2.78. Note that the highest probabilities are at x = 2 and x
= 3 which are near the mean. The probability is slightly higher at x = 2 than at x = 3 even
though x = 3 is nearer to the mean because of the “piling up” effect of x = 0.
5.62 This is a binomial distribution with n = 22 and p = .64.
The mean is n⋅p = 22(.64) = 14.08 and the standard deviation is:
σ = )36)(.64(.22=⋅⋅ qpn = 2.25
43. Chapter 5: Discrete Distributions 43
The x value with the highest peak on the graph is at x = 14 followed by x = 15
and x = 13 which are nearest to the mean.
5.63 This is the graph of a Poisson Distribution with λ = 1.784. Note the high
probabilities at x = 1 and x = 2 which are nearest to the mean. Note also that the
probabilities for values of x > 8 are near to zero because they are so far away
from the mean or expected value.
