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5-1 Copyright ©2015 Pearson Education, Inc.
CHAPTER 5 Discrete Probability Distributions
5.1 ( ) ( )5 1.0 0.06 0.11 0.24 0.27 0.20 1.0 0.88 0.12P x = = − + + + + == − =
5.2 a) µ = (1)(0.18) + (2)(0.25) + (3)(0.35) + (4)(0.22) = 2.61
b) 27.850 (2.61)σ = − = 1.019
5.3 a) µ = (10)(0.10) + (15)(0.30) + (20)(0.20) + (25)(0.30) + (30)(0.10) = 20.0
b) 2435 (20.0)σ = − = 5.916
5.4 A – not a discrete probability distribution (the sum of the probabilities of all outcomes is greater than 1) B – discrete probability distribution C – discrete probability distribution D – not a discrete probability distribution (probability of one outcome is greater than 1) A – not a discrete probability distribution (the sum of the probabilities of all outcomes is less than 1) 5.5
a) µ = 88 40 42 20 7 3
(0) (1) (2) (3) (4) (5)200 200 200 200 200 200
+ + + + + =
= (0)(0.44) + (1)(0.20) + (2)(0.21) + (3)(0.10) + (4)(0.035) + (5)(0.015) = 1.135
b) 22.875 (1.135)σ = − = 1.260
5.6 a) Stanton:
( ) ( ) ( ) ( ) ( )
4 12 8 36 20(1) (2) (3) (4) (5)
80 80 80 80 80
(1) 0.05 (2) 0.15 (3) 0.10 (4) 0.45 (5) 0.25 3.7
μ = + + + + =
= + + + + =
Newark:
( ) ( ) ( ) ( ) ( )
12 12 18 15 18(1) (2) (3) (4) (5)
75 75 75 75 75
(1) 0.16 (2) 0.16 (3) 0.24 (4) 0.20 (5) 0.24 3.2
μ = + + + + =
= + + + + =
5-2 Chapter 5
Copyright ©2015 Pearson Education, Inc.
b) Stanton: 215.00 (3.7)σ = − = 1.14; Newark: 212.16 (3.2)σ = − =1.36
c) Stanton’s fast-food restaurants have higher average customer satisfaction rating than ones in Newark. Customer satisfaction ratings for Stanton’s fast-food restaurants are more consistent when compared with ones in Newark. 5.7 a) µ = (3)(0.23) + (4)(0.57) + (5)(0.14) + (6)(0.06) = 4.03
b) 216.850 (4.03)σ = − = 0.78
5.8 EMV = ($4,000)(0.30) + ($20,000)(0.45) + ($36,000)(0.25) = $19,200 5.9 Profit = Revenue– Costs Revenue is based on the number loaves sold, which is the minimum of the supply of loaves and the eventual demand.
Demand Profit (Bake 25 loaves) Probability 25 (25)($6) – (25)($3) = $75 0.35 50 (25)($6) – (25)($3) = $75 0.25 75 (25)($6) – (25)($3) = $75 0.40
EMV (25) = ($75)(0.35) + ($75)(0.25) + ( $75)(0.40) = $75
Demand Profit (Bake 50 loaves) Probability
25 (25)($6) – (50)($3) = $0 0.35 50 (50)($6) – (50)($3) = $150 0.25 75 (50)($6) – (50)($3) = $150 0.40
EMV (50) = ($0)(0.35) + ($150)(0.25) + ( $150)(0.40) = $97.50
Demand Profit (Bake 75 loaves) Probability
25 (25)($6) – (75)($3) = -$75 0.35 50 (50)($6) – (75)($3) = $75 0.25 75 (75)($6) – (75)($3) = $225 0.40
EMV (75) = (-$75)(0.35) + ($75)(0.25) + ( $225)(0.40) = $82.50 The most profitable is to bake 50 loaves every morning.
Discrete Probability Distributions 5-3
Copyright ©2015 Pearson Education, Inc.
5.10
a) 2 57!(2,7) (0.6) (0.4) 0.0774
(7 2)!2!P = =
−
b) 0 7 1 67! 7!(0,7) (1,7) (0.60) (0.40) (0.60) (0.40) 0.0188
(7 0)!0! (7 1)!1!P P+ = + =
− −
c) 6 1 7 07! 7!(6,7) (7,7) (0.60) (0.40) (0.60) (0.40) 0.1586
(7 6)!6! (7 7)!7!P P+ = + =
− −
5.11
a) 3 58!(3,8) (0.35) (0.65) 0.2786
(8 3)!3!P = =
−
b)
( )
0 8
1 7
2 6
8!(0,8) (0.35) (0.65) 0.0319
(8 0)!0!
8!(1,8) (0.35) (0.65) 0.1373
(8 1)!1!
8!(2,8) (0.35) (0.65) 0.2587
(8 2)!2!
3 0.0319 0.1373 0.2587 0.4279
P
P
P
P x
= =−
= =−
= =−
< = + + =
c)
( )
6 2
7 1
8 0
8!(6,8) (0.35) (0.65) 0.0217
(8 6)!6!
8!(7,8) (0.35) (0.65) 0.0033
(8 7)!7!
8!(8,8) (0.35) (0.65) 0.0002
(8 8)!8!
6 0.0217 0.0033 0.0002 0.0252
P
P
P
P x
= =−
= =−
= =−
≥ = + + =
5.12
a) 4 15!(4,5) (0.25) (0.75) 0.0146
(5 4)!4!P = =
−
b) 4 37!(4,7) (0.25) (0.75) 0.0577
(7 4)!4!P = =
−
c) 4 610!(4,10) (0.25) (0.75) 0.1460
(10 4)!4!P = =
−
5-4 Chapter 5
Copyright ©2015 Pearson Education, Inc.
5.13 (15)(0.65) 9.75; (15)(0.65)(0.35) 1.847μ σ= = = =
5.14 ( 14) 0.1091 0.1746 0.2182 0.2054 0.1369 0.0576 0.0115 0.9133P x ≥ = + + + + + + =
5.15
a) 3 710!(3,10) (0.28) (0.72) 0.2642
(10 3)!3!P = =
−
b)
( )
0 10
1 9
2 8
3 7
10!(0,10) (0.28) (0.72) 0.0374
(10 0)!0!
10!(1,10) (0.28) (0.72) 0.1456
(10 1)!1!
10!(2,10) (0.28) (0.72) 0.2548
(10 2)!2!
10!(3,10) (0.28) (0.72) 0.2642
(10 3)!3!
3 0.0374 0.1456 0.2548 0.2642 0.
P
P
P
P
P x
= =−
= =−
= =−
= =−
≤ = + + + = 7020
c)
0 10
1 9
2 8
3 7
4 6
10!(0,10) (0.28) (0.72) 0.0374
(10 0)!0!
10!(1,10) (0.28) (0.72) 0.1456
(10 1)!1!
10!(2,10) (0.28) (0.72) 0.2548
(10 2)!2!
10!(3,10) (0.28) (0.72) 0.2642
(10 3)!3!
10!(4,10) (0.28) (0.72)
(10 4)!4!
P
P
P
P
P
= =−
= =−
= =−
= =−
= =−
( ) ( )( )
0.1798
5 1 5
5 1 0.0374 0.1456 0.2548 0.2642 0.1798 0.1182
P x P x
P x
≥ = − <
≥ = − − − − − =
d) (10)(0.28) 2.8; (10)(0.28)(0.72) 1.42μ σ= = = =
Discrete Probability Distributions 5-5
Copyright ©2015 Pearson Education, Inc.
e)
5.16
a) 7 18!(7,8) (0.90) (0.10) 0.3826
(8 7)!7!P = =
−
b) ( ) 8 08!7 (8,8) (0.90) (0.10) 0.4305
(8 8)!8!P x P> = = =
−
c)
( )
( )
6 2
7 1
8 0
8!(6,8) (0.90) (0.10) 0.1488
(8 6)!6!
8!(7,8) (0.90) (0.10) 0.3826
(8 7)!7!
8!(8,8) (0.90) (0.10) 0.4305
(8 8)!8!
6
6
( 6) 1
1 1 0.1488 0.3826 0.4305 0.0381
P
P
P
x
x
P x PP
= =−
= =−
= =−
≥
≥
< = −− = − − − =
5-6 Chapter 5
Copyright ©2015 Pearson Education, Inc.
d) 4 48!(4,8) (0.90) (0.10) 0.0046
(8 4)!4!P = =
−
The probability of 4 out of 8 customers being satisfied is very low assuming that 90% of the customer base is satisfied. Based on this sample, it is not likely that 90% of the customers are satisfied. 5.17
a) 2 911!(2,11) (0.25) (0.75) 0.2581
(11 2)!2!P = =
−
b)
( )
0 11
1 10
2 9
11!(0,11) (0.25) (0.75) 0.0422
(11 0)!0!
11!(1,11) (0.25) (0.75) 0.1549
(11 1)!1!
11!(2,11) (0.25) (0.75) 0.2581
(11 2)!2!
3 0.0422 0.1549 0.2581 0.4552
P
P
P
P x
= =−
= =−
= =−
< = + + =
c)
0 11
1 10
2 9
3 8
4 7
11!(0,11) (0.25) (0.75) 0.0422
(11 0)!0!
11!(1,11) (0.25) (0.75) 0.1549
(11 1)!1!
11!(2,11) (0.25) (0.75) 0.2581
(11 2)!2!
11!(3,11) (0.25) (0.75) 0.2581
(11 3)!3!
11!(4,11) (0.25) (0.75)
(11 4)!4!
P
P
P
P
P
= =−
= =−
= =−
= =−
=−
( )
5 6
0.1721
11!(5,11) (0.25) (0.75) 0.0803
(11 5)!5!
5 1.0 0.0422 0.1549 0.2581 0.2581 0.1721 0.0803 0.0343
P
P x
=
= =−
> = − − − − − − =
d) (11)(0.25) 2.75; (11)(0.25)(0.75) 1.436μ σ= = = =
Discrete Probability Distributions 5-7
Copyright ©2015 Pearson Education, Inc.
e)
5.18
a) 3 69!(3,9) (0.35) (0.65) 0.2716
(9 3)!3!P = =
−
b)
( )
0 9
1 8
2 7
3 6
9!(0,9) (0.35) (0.65) 0.0207
(9 0)!0!
9!(1,9) (0.35) (0.65) 0.1004
(9 1)!1!
9!(2,9) (0.35) (0.65) 0.2162
(9 2)!2!
9!(3,9) (0.35) (0.65) 0.2716
(9 3)!3!
4 0.0207 0.1004 0.2162 0.2716 0.6089
P
P
P
P
P x
= =−
= =−
= =−
= =−
< = + + + =
c)
6 3
7 2
9!(6,9) (0.35) (0.65) 0.0424
(9 6)!6!
9!(7,9) (0.35) (0.65) 0.0098
(9 7)!7!
(6,9) (7,9) 0.0424 0.0098 0.0522
P
P
P P
= =−
= =−
+ = + =
5-8 Chapter 5
Copyright ©2015 Pearson Education, Inc.
d) (9)(0.35) 3.15; (9)(0.35)(0.65) 1.43μ σ= = = =
e)
5.19 a)
1 1112!
(1,12) (0.14) (0.86) 0.3197(12 1)!1!
P = =−
b)
( )
0 12
1 11
2 10
3 9
12!(0,12) (0.14) (0.86) 0.1637
(12 0)!0!
12!(1,12) (0.14) (0.86) 0.3197
(12 1)!1!
12!(2,12) (0.14) (0.86) 0.2863
(12 2)!2!
12!(3,12) (0.14) (0.86) 0.1553
(12 3)!3!
4 0.1637 0.3197 0.2863 0.1553
P
P
P
P
P x
= =−
= =−
= =−
= =−
< = + + + = 0.9250
Discrete Probability Distributions 5-9
Copyright ©2015 Pearson Education, Inc.
c)
0 12
1 11
2 10
12!(0,12) (0.14) (0.86) 0.1637
(12 0)!0!
12!(1,12) (0.14) (0.86) 0.3197
(12 1)!1!
12!(2,12) (0.14) (0.86) 0.2863
(12 2)!2!
( 2) 1 0.1637 0.3197 0.2863 0.2303
P
P
P
P x
= =−
= =−
= =−
> = − − − =
d) (12)(0.14) 1.68; (12)(0.14)(0.86) 1.202μ σ= = = = e)
5.20
a) 0 1515!(0,15) (0.06) (0.94) 0.3953
(15 0)!0!P = =
−
b)
( )
0 15
1 14
2 13
15!(0,15) (0.06) (0.94) 0.3953
(15 0)!0!
15!(1,15) (0.06) (0.94) 0.3785
(15 1)!1!
15!(2,15) (0.06) (0.94) 0.1691
(15 2)!2!
3 0.3953 0.3785 0.1691 0.9429
P
P
P
P x
= =−
= =−
= =−
< = + + =
5-10 Chapter 5
Copyright ©2015 Pearson Education, Inc.
c)
0 15
1 14
15!(0,15) (0.06) (0.94) 0.3953
(15 0)!0!
15!(1,15) (0.06) (0.94) 0.3785
(15 1)!1!
( 1) 1 0.3953 0.3785 0.2262
P
P
P x
= =−
= =−
> = − − =
d)
e) 4 1115!(4,15) (0.06) (0.94) 0.0090
(15 4)!4!P = =
−
The probability of 4 out of 15 customers making a purchase on the web site is very low assuming that 6% of the customers make a purchase. Based on this sample, it is not likely that 6% of the customers make purchases. 5.21
a) 9 514!(9,14) (0.70) (0.30) 0.1963
(14 9)!9!P = =
−
b) 14 014!(14,14) (0.70) (0.30) 0.0068
(14 14)!14!P = =
−
Discrete Probability Distributions 5-11
Copyright ©2015 Pearson Education, Inc.
c)
12 2
13 1
14 0
14!(12,14) (0.70) (0.30) 0.1134
(14 12)!12!
14!(13,14) (0.70) (0.30) 0.0407
(14 13)!13!
14!(14,14) (0.70) (0.30) 0.0068
(14 14)!14!
( 11) 1 0.1134 0.0407 0.0068 0.8391
P
P
P
P x
= =−
= =−
= =−
≤ = − − − =
d) .715(14)(0.70) 9.8; (14)(0.70)(0.30) 1μ σ= = = = e)
5.22
a) ( )( )3 3.03.0 2.71828
(3) 0.22403!
P−
= =
5-12 Chapter 5
Copyright ©2015 Pearson Education, Inc.
b)
( )( )
( )( )
( )( )
0 3.0
1 3.0
2 3.0
3.0 2.71828(0) 0.0498
0!
3.0 2.71828(1) 0.1494
1!
3.0 2.71828(2) 0.2240
2!( 2) 0.0498 0.1494 0.2240 0.4232
P
P
P
P x
−
−
−
= =
= =
= =
≤ = + + =
c)
( )( )
( )( )
( )( )
( )( )
( ) ( )( )
( ) ( )( )
0 3.0
1 3.0
2 3.0
4 3.0
3 3.0
3.0 2.71828(0) 0.0498
0!
3.0 2.71828(1) 0.1494
1!
3.0 2.71828(2) 0.2240
2!
(3) 0.2240
3.0 2.718284 0.1680
4!4 4
4
3.0 2.71828
3!
1
1 0.0498 0.1494 0.2240 0.2240 0.1680 0.18
P
P
P
P
P
x x
x
P PP
−
−
−
−
−
= =
= =
= =
= =
= =
> = ≤
>
−= − − − − − = 48
5.23
a) ( )( )5 4.74.7 2.71878
(5) 0.17385!
P−
= =
Discrete Probability Distributions 5-13
Copyright ©2015 Pearson Education, Inc.
b)
( )( )
( )( )
( )( )
( )( )
( )( )
( )( )
( )( )
0 4.7
1 4.7
2 4.7
3 4.7
4 4.7
5 4.7
6 4.7
4.7 2.71878(0) 0.0091
0!
4.7 2.71878(1) 0.0427
1!
4.7 2.71878(2) 0.1005
2!
4.7 2.71878(3) 0.1574
3!
4.7 2.71878(4) 0.1849
4!
4.7 2.71878(5) 0.1738
5!
4.7 2.71878(6) 0.1
6!
P
P
P
P
P
P
P
−
−
−
−
−
−
−
= =
= =
= =
= =
= =
= =
= =
( )
362
6
1 0.0091 0.0427 0.1005 0.1574 0.1849 0.1738 0.1362
( 6) 1
( 6)
( 6) 0.1954
xP x PP xP x
≤= − − − − − − −
> = −>> =
c)
( )( )
( )( )
( )( )
( )( )
0 4.7
1 4.7
2 4.7
3 4.7
4.7 2.71878(0) 0.0091
0!
4.7 2.71878(1) 0.0427
1!
4.7 2.71878(2) 0.1005
2!
4.7 2.71878(3) 0.1574
3!0.0091 0.0427 0.1005 0.1574( 3) 0.3097
P
P
P
P
P x
−
−
−
−
= =
= =
= =
= =
+ + +≤ = =
5.24
a) ( )( )4 2.02.0 2.71828
(4) 0.09024!
P−
= =
b) ( )( )4 3.03.0 2.71828
(4) 0.16804!
P−
= =
5-14 Chapter 5
Copyright ©2015 Pearson Education, Inc.
c) ( )( )4 4.04.0 2.71828
(4) 0.19544!
P−
= =
d) As the value of λ approached 4.0, the probability that 4x = increases.
5.25 . 8.5 2.928 5; μ λ σ == = =
5.26
a) ( )( )0 2.62.6 2.71828
(0) 0.07430!
P−
= =
b)
( )( )
( )( )
( )( )
( )( )
( )
0 2.6
1 2.6
2 2.6
3 2.6
2.6 2.71828(0) 0.0743
0!
2.6 2.71828(1) 0.1931
1!
2.6 2.71828(2) 0.2510
2!
2.6 2.71828(3) 0.2176
3!4 0.0743 0.1931 0.2510 0.2176 0.7360
P
P
P
P
P x
−
−
−
−
= =
= =
= =
= =
< = + + + =
c)
( )( )
( )( )
( )
0 2.6
1 2.6
2.6 2.71828(0) 0.0743
0!
2.6 2.71828(1) 0.1931
1!( 1) 1 1 1 0.0743 0.1931 0.7326
P
P
P x P x
−
−
= =
= =
> = − ≤ = − − =
5.27
a) ( )( )5 2.12.1 2.71828
(5) 0.04175!
P−
= =
b) ( )( )0 2.12.1 2.71828
(0) 0.12250!
P−
= =
Discrete Probability Distributions 5-15
Copyright ©2015 Pearson Education, Inc.
c)
( )
( ) ( )( )
( ) ( )( )
( ) ( )( )
0 2.1
1 2.1
2 2.1
( 2) 1 2
2.1 2.718280 0.1225
0!
2.1 2.718281 0.2572
1!
2.1 2.718282 0.2700
2!( 2) 1 0.1225 0.2572 0.2700 0.3503
P x P x
P
P
P
P x
−
−
−
> = − ≤
= =
= =
= =
> = − − − =
d) 2.1 1.45σ ==
5.28
a) ( )( )7 4.54.5 2.71828
(7) 0.08247!
P−
= =
b)
( )( )
( )( )
( )( )
( )
0 4.5
1 4.5
2 4.5
4.5 2.71828(0) 0.0111
0!
4.5 2.71828(1) 0.0500
1!
4.5 2.71828(2) 0.1125
2!3 0.0111 0.0500 0.1125 0.1736
P
P
P
P x
−
−
−
= =
= =
= =
< = + + =
c)
( )( )
( )( )
( )( )
( )( )
( )( )
( )
0 4.5
1 4.5
2 4.5
3 4.5
4 4.5
4.5 2.71828(0) 0.0111
0!
4.5 2.71828(1) 0.0500
1!
4.5 2.71828(2) 0.1125
2!
4.5 2.71828(3) 0.1687
3!
4.5 2.71828(4) 0.1898
4!( 5) 1 5
( 5) 1 0.0111 0.0500 0.1125 0.1687 0.189
P
P
P
P
P
P x P xP x
−
−
−
−
−
= =
= =
= =
= =
= =
≥ = − <≥ = − − − − − 8 0.4679=
5-16 Chapter 5
Copyright ©2015 Pearson Education, Inc.
5.29
a) ( )( )0 1.51.5 2.71828
(0) 0.22310!
P−
= =
b)
( ) ( )( )
( ) ( )( )
( ) ( )( )
( )
0 1.5
1 1.5
2 1.5
1.5 2.718280 0.2231
0!
1.5 2.718281 0.3347
1!
1.5 2.718282 0.2510
2!( 3) 1 3 1 0.2231 0.3347 0.2510 0.1912
P
P
P
P x P x
−
−
−
= =
= =
= =
≥ = − < = − − − =
c) ( )( )3 3.03.0 2.71828
(3) 0.22403!
P−
= =
d)
( )( )
( )( )
( )( )
( )
0 3.0
1 3.0
2 3.0
3.0 2.71828(0) 0.0498
0!
3.0 2.71828(1) 0.1494
1!
3.0 2.71828(2) 0.2240
2!2 0.0498 0.1494 0.2240 0.4232
P
P
P
P x
−
−
−
= =
= =
= =
≤ = + + =
5.30
a) 2 2022!(2,22) (0.015) (0.985) 0.0384
(22 2)!2!P = =
−
b) 2 (22)(0.015)((22)(0.015)) 2.71828
(2) 0.03912!
P−
= =
c) The probabilities are close enough, but fewer calculations were needed for the result when using the Poisson distribution
Discrete Probability Distributions 5-17
Copyright ©2015 Pearson Education, Inc.
5.31 a)
( )
0 25
1 24
2 23
25!(0,25) (0.016) (0.984) 0.6682
(25 0)!0!
25!(1, 25) (0.016) (0.984) 0.2716
(25 1)!1!
25!(2,25) (0.016) (0.984) 0.0530
(25 2)!2!
3 0.6682 0.2716 0.0530 0.9928
P
P
P
P x
= =−
= =−
= =−
< = + + =
b)
( )( )
( )
0 (0.40)
1 (0.40)
2 (0.40)
25 0.016 0.40
(0.40) 2.71828(0) 0.6703
0!
(0.40) 2.71828(1) 0.2681
1!
(0.40) 2.71828(2) 0.0536
2!3 0.6703 0.2681 0.0536 0.9920
np
P
P
P
P x
−
−
−
= =
= =
= =
= =
< = + + =
5.32
a) ( ) 12 6 5 3 6 3
12 5
(3) 0.3788C CP x P
C− −= = =
b) 12 6 5 2 6 2
12 5
( ) (2) 0.3788C CP x P
C− −= = =
c) 12 6 5 0 6 0 12 6 5 1 6 1
12 5 12 5
( ) (0) (1) 0.1212C C C CP x P P
C C− − − −= + = + =
d) ( )( )2
12 5(5)(6) (5)(6)(12 6)2.5; 0.89
12 12 12 1μ σ −−= =
−= =
5.33
a) 9 5 3 0 5 0
9 3
( ) (0) 0.0476C CP x P
C− −= = =
b) ( ) 9 5 3 0 5 0 9 5 3 1 5 1
9 3 9 3
( ) 1 (0) (1) 1 0.5952C C C CP x P P
C C− − − −
= − + = − + =
c) 9 5 3 0 5 0 9 5 3 1 5 1 9 5 3 2 5 2
9 3 9 3 9 3
( ) (0) (1) (2) 0.8810C C C C C CP x P P P
C C C− − − − − −= + + = + + =
5-18 Chapter 5
Copyright ©2015 Pearson Education, Inc.
d) ( )( )2
9 3(3)(5) (3)(5)(9 5)1.67; 0.7453
9 9 9 1μ σ −−= =
−= =
5.34 14, 6, 4, 3;N R n x= = = = 14 6 4 3 6 3
14 4
( ) (3) 0.1598C CP x P
C− −= = =
5.35 18, 10, 6, 6;N R n x= = = = 18 10 6 6 10 6
18 6
( ) (6) 0.0113C CP x P
C− −= = =
5.36
a) 9, 2, 2, 0;N R n x= = = = 9 2 2 0 2 0
9 2
( ) (0) 0.5833C CP x P
C− −= = =
b) 9, 2, 2, 1;N R n x= = = = 9 2 2 1 2 1
9 2
( ) (1) 0.3889C CP x P
C− −= = =
c) 9, 2, 2, 2;N R n x= = = = 9 2 2 2 2 2
9 2
( ) (2) 0.0278C CP x P
C− −= = =
d) ( )( )2
9 2(2)(2) (2)(2)(9 2)0.444; 0.55
9 9 9 1μ σ −−= =
−= =
e) HYPGEOM.DIST(0, 2, 2, 9, FALSE) = 0.5833 HYPGEOM.DIST(1, 2, 2, 9, FALSE) = 0.3889 HYPGEOM.DIST(2, 2, 2, 9, FALSE) = 0.0278
5.37
a) 12, 5, 6, 4;N R n x= = = = 12 5 6 4 5 4
12 6
( ) (4) 0.1136C CP x P
C− −= = =
b) 12, 5, 6, 2;N R n x= = = = 12 5 6 2 5 2
12 6
( ) (2) 0.3788C CP x P
C− −= = =
c) 12, 5, 6, 3;N R n x= = = = 12 5 6 3 5 3
12 6
( ) (3) 0.3788C CP x P
C− −= = =
d) ( )( )2
12 6(6)(5) (6)(5)(12 5)2.50; 0.8919
12 12 12 1μ σ −−= =
−= =
e) HYPGEOM.DIST(4, 6, 5, 12, FALSE) = 0.1136 HYPGEOM.DIST(2, 6, 5, 12, FALSE) = 0.3788 HYPGEOM.DIST(3, 6, 5, 12, FALSE) = 0.3788
Discrete Probability Distributions 5-19
Copyright ©2015 Pearson Education, Inc.
5.38
a) 20, 8, 3, 3;N R n x= = = = 20 8 3 3 8 3
20 3
( ) (3) 0.0491C CP x P
C− −= = =
b) 20, 8, 3, 1;N R n x= = = = 20 8 3 1 8 1
20 3
( ) (1) 0.4632C CP x P
C− −= = =
c) 20, 8, 3, 2;N R n x= = = = 20 8 3 2 8 2
20 3
( ) (2) 0.2947C CP x P
C− −= = =
d) 20, 8, 3, 0;N R n x= = = = 20 8 3 0 8 0
20 3
( ) (0) 0.1930C CP x P
C− −= = =
e) ( )( )2
20 3(3)(8) (3)(8)(20 8)1.2; 0.80
20 20 20 1μ σ −−= =
−= =
f) HYPGEOM.DIST(3, 3, 8, 20, FALSE) = 0.0491 HYPGEOM.DIST(1, 3, 8, 20, FALSE) = 0.4632 HYPGEOM.DIST(2, 3, 8, 20, FALSE) = 0.2947 HYPGEOM.DIST(0, 3, 8, 20, FALSE) = 0.1930
5.39 1 0.10 45 2 0.20 90 3 0.50 225 4 0.20 90 5.40 a) µ = (0)(0.23) + (1)(0.32) + (2)(0.22) + (3)(0.15) + (4)(0.08) = 1.53
b) 23.83 (1.53)σ = − = 1.22
5.41 a) µ = (0)(0.37) + (1)(0.28) + (2)(0.21) + (3)(0.10) + (4)(0.04) = 1.16
b) 2 1.152.66 (1.16)σ == −
5.42 a) 8 AM class: µ = 2.58; 10 AM class: µ = 1.79 b) 8 AM class: 1.39σ = ; 10 AM class: 1.33σ = c) These results were expected 5.43 EMV = ($10,000)(0.03) + ($0)(0.97) = $300. The cost of insurance is $100 greater than EMV, so it should not be purchased.
5-20 Chapter 5
Copyright ©2015 Pearson Education, Inc.
5.44 Portfolio A: EMV = (-$1,500)(0.25) + ($2,500)(0.60) + ($4,000)(0.15) = $1,725 Portfolio B: EMV = (-$4,000)(0.25) + ($3,000)(0.60) + ($6,000)(0.15) = $1,700 Investor should choose the portfolio A. 5.45 Profit = $16 revenue + $5 revenue – cost to purchase shirts The $16 revenue is based on the number of shirts sold at $16, which is the minimum of the supply of shirts (3000) and the eventual demand. Demand Profit (3,000 shirts) Probability 1,000 (1,000)($16) + (2,000)($5) – (3,000)($10) = -$4,000 0.15 2,000 (2,000)($16) + (1,000)($5) – (3,000)($10) = $7,000 0.25 3,000 (3,000)($16) + (0)($5) – (3,000)($10) = $18,000 0.30 4,000 (3,000)($16) + (0)($5) – (3,000)($10) = $18,000 0.30 EMV = (-$4,000)(0.15) + ($7,000)(0.25) + ($18,000)(0.30) + ($18,000)(0.30) = $11,950 5.46 Profit = $14 revenue + $2 revenue – cost to purchase calendars The $14 revenue is based on the number of calendars sold at $14, which is the minimum of the supply of calendars and the eventual demand. Demand Profit (Order 100 calendars) Probability 100 (100)($14) + (0)($2) – (100)($6) = $800 0.25 200 (100)($14) + (0)($2) – (100)($6) = $800 0.40 300 (100)($14) + (0)($2) – (100)($6) = $800 0.35 EMV = ($800)(0.25) + ($800)(0.40) + ($800)(0.35) = $800 Demand Profit (Order 200 calendars) Probability 100 (100)($14) + (100)($2) – (200)($6) = $400 0.25 200 (200)($14) + (0)($2) – (200)($6) = $1,600 0.40 300 (200)($14) + (0)($2) – (200)($6) = $1,600 0.35 EMV = ($400)(0.25) + ($1,600)(0.40) + ($1,600)(0.35) = $1,300 Demand Profit (Order 300 calendars) Probability 100 (100)($14) + (200)($2) – (300)($6) = $0 0.25 200 (200)($14) + (100)($2) – (300)($6) = $1,200 0.40 300 (300)($14) + (0)($2) – (300)($6) = $2,400 0.35 EMV = ($0)(0.25) + ($1,200)(0.40) + ($2,400)(0.35) = $1,320
Discrete Probability Distributions 5-21
Copyright ©2015 Pearson Education, Inc.
Bob’s Bookstore should order 300 calendars.
5.47 14 014!(14,14) (0.5) (0.5) 0.000061
(14 14)!14!P = =
−
5.48
a) 6 915!(6,15) (0.45) (0.55) 0.1914
(15 6)!6!P = =
−
b) 9 615!(9,15) (0.45) (0.55) 0.1048
(15 9)!9!P = =
− c)
( )
4 11
5 10
15!(4,15) (0.45) (0.55) 0.0780
(15 4)!4!
15!(5,15) (0.45) (0.55) 0.1404
(15 5)!5!
4 or 5 0.0780 0.1404 0.2184
P
P
P x x
= =−
= =−
= = = + =
d) (15)(0.45) 6.75; (15)(0.45)(0.55) 1.926μ σ= = = =
e)
5-22 Chapter 5
Copyright ©2015 Pearson Education, Inc.
5.49
a) 3 25!(3,5) (0.24) (0.76) 0.0798
(5 3)!3!P = =
−
b) 0 55!(0,5) (0.24) (0.76) 0.2536
(5 0)!0!P = =
−
c)
0 5
1 4
2 3
5!(0,5) (0.24) (0.76) 0.2536
(5 0)!0!
5!(1,5) (0.24) (0.76) 0.4003
(5 1)!1!
5!(2,5) (0.24) (0.76) 0.2529
(5 2)!2!
( 2) 0.2536 0.4003 0.2529 0.9068
P
P
P
P x
= =−
= =−
= =−
≤ = + + =
d)
5.50
a) 4 37!(4,7) (0.37) (0.63) 0.1640
(7 4)!4!P = =
−
Discrete Probability Distributions 5-23
Copyright ©2015 Pearson Education, Inc.
b)
0 7
1 6
2 5
7!(0,7) (0.37) (0.63) 0.0394
(7 0)!0!
7!(1,7) (0.37) (0.63) 0.1619
(7 1)!1!
7!(2,7) (0.37) (0.63) 0.2853
(7 2)!2!
( 2) 1.0 0.0394 0.1619 0.2853 0.5134
P
P
P
P x
= =−
= =−
= =−
> = − − − =
c) (7)(0.37) 2.59; (7)(0.37)(0.63) 1.28μ σ= = = = d)
5.51
a) 10 010!(10,10) (0.80) (0.20) 0.1074
(10 10)!10!P = =
−
b)
8 2
9 1 10 0
10!(8,10) (9,10) (10,10) (0.80) (0.20)
(10 8)!8!
10! 10!(0.80) (0.20) (0.80) (0.20) 0.6778
(10 9)!9! (10 10)!10!
P P P+ + = +−
+ =− −
c) 6 4 7 310! 10!(6,10) (7,10) (0.80) (0.20) (0.80) (0.20) 0.2894
(10 6)!6! (10 7)!7!P P+ = + =
− −
5-24 Chapter 5
Copyright ©2015 Pearson Education, Inc.
d)
e) 5 510!(5,10) (0.8) (0.2) 0.0264
(10 5)!5!P = =
−
The probability of 5 out of 10 rounds being below 90 is very low assuming that 80% of the rounds are below 90. Based on this sample, it is not likely that 80% of the rounds are below 90. 5.52
a) 12 012!(12,12) (0.79) (0.21) 0.0591
(12 12)!12!P = =
−
b)
( )
10 2
11 1
12 0
12!(10,12) (0.79) (0.21) 0.2756
(12 10)!10!
12!(11,12) (0.79) (0.21) 0.1885
(12 11)!11!
12!(12,12) (0.79) (0.21) 0.0591
(12 12)!12!
9 0.2756 0.1185 0.0591 0.5232
P
P
P
P x
= =−
= =−
= =−
> = + + =
Discrete Probability Distributions 5-25
Copyright ©2015 Pearson Education, Inc.
c)
8 4
9 3
10 2
11 1
12!(8,12) (0.79) (0.21) 0.1460
(12 8)!8!
12!(9,12) (0.79) (0.21) 0.2442
(12 9)!9!
12!(10,12) (0.79) (0.21) 0.2756
(12 10)!10!
12!(11,12) (0.79) (0.21) 0.1885
(12 11)!11!
12!(12,12) (0.79
(12 12)!12!
P
P
P
P
P
= =−
= =−
= =−
= =−
=−
( )
12 0) (0.21) 0.0591
7 1 0.1460 0.2442 0.2756 0.1885 0.0591 0.0866P x
=
≤ = − − − − − =
d) (12)(0.79) 9.48; (12)(0.79)(0.21) 1.411μ σ= = = = e)
5.53
a) 12 113!(12,13) (0.70) (0.30) 0.0540
(13 12)!12!P = =
−
5-26 Chapter 5
Copyright ©2015 Pearson Education, Inc.
b)
11 2
12 1 13 0
13!(11,13) (12,13) (13,13) (0.70) (0.30)
(13 11)!11!
13! 13!(0.70) (0.30) (0.70) (0.30) 0.2025
(13 12)!12! (13 13)!13!
P P P+ + = +−
+ =− −
c)
( )12 1 13 0
( ) 1 (12,13) (13,13)
13! 13!1 (0.70) (0.30) (0.70) (0.30) 0.9363
(13 12)!12! (13 13)!13!
P x P P= − + =
− + = − −
d)
e) 5 813!(5,13) (0.7) (0.3) 0.0142
(13 5)!5!P = =
−
The probability that 5 out of 13 men did not use their shoes on the basketball court is very low assuming that 70% of men do not wear them on courts. Based on this sample, it is not likely that 70% of men do not wear their basketball shoes on the basketball court. 5.54
a) 6 06!(6,6) (0.86) (0.14) 0.4046
(6 6)!6!P = =
−
Discrete Probability Distributions 5-27
Copyright ©2015 Pearson Education, Inc.
b)
4 2
5 1
6 0
6!(4,6) (0.86) (0.14) 0.1608
(6 4)!4!
6!(5,6) (0.86) (0.14) 0.3952
(6 5)!5!
6!(6,6) (0.86) (0.14) 0.4046
(6 6)!6!
( 4) 0.1608 0.3952 0.4046 0.9606
P
P
P
P x
= =−
= =−
= =−
≥ = + + =
c)
5 1
6 0
6!(5,6) (0.86) (0.14) 0.3952
(6 5)!5!
6!(6,6) (0.86) (0.14) 0.4046
(6 6)!6!
( 5) 1 0.3952 0.4046 0.2002
P
P
P x
= =−
= =−
< = − − =
d) (6)(0.86) 5.16; (6)(0.86)(0.14) 0.850μ σ= = = = e)
5.55
a) 9 110!(9,10) (0.79) (0.21) 0.2517
(10 9)!9!P = =
−
5-28 Chapter 5
Copyright ©2015 Pearson Education, Inc.
b)
8 2
9 1
10 0
10!(8,10) (0.79) (0.21) 0.3011
(10 8)!8!
10!(9,10) (0.79) (0.21) 0.2517
(10 9)!9!
10!(10,10) (0.79) (0.21) 0.0947
(10 10)!10!
( 7) 0.3011 0.2517 0.0947 0.6475
P
P
P
P x
= =−
= =−
= =−
> = + + =
c)
10 010!
(10,10) (0.79) (0.21) 0.0947(10 10)!10!
( 10) 1 0.0947 0.9053
P
P x
= =−
< = − =
d) (10)(0.79) 7.9; (10)(0.79)(0.21) 1.29μ σ= = = =
e
5.56 a) (12,20) 0.0727P =
b)
(13, 20) (14, 20) (15, 20) (16, 20)
(17, 20) (18, 20) (19, 20) (20, 20) 0.0580
P P P PP P P P
+ + + ++ + + + =
Discrete Probability Distributions 5-29
Copyright ©2015 Pearson Education, Inc.
c)
(0, 20) (1, 20) (2, 20) (3, 20) (4, 20)
(5, 20) (6, 20) (7, 20) (8, 20) (9, 20) 0.5914
P P P P PP P P P P
+ + + + ++ + + + + =
d) (20)(0.45) 9; (20)(0.45)(0.55) 2.2μ σ= = = =
e)
5.57
a) 2 79!(2,9) (0.23) (0.77) 0.3056
(9 2)!2!P = =
−
b)
0 9
1 8
2 7
3 6
9!(0,9) (0.23) (0.77) 0.0952
(9 0)!0!
9!(1,9) (0.23) (0.77) 0.2558
(9 1)!1!
9!(2,9) (0.23) (0.77) 0.3056
(9 2)!2!
9!(3,9) (0.23) (0.77) 0.2130
(9 3)!3!
( 4) 0.0952 0.2558 0.3056 0.2130 0.8696
P
P
P
P
P x
= =−
= =−
= =−
= =−
< = + + + =
5-30 Chapter 5
Copyright ©2015 Pearson Education, Inc.
c)
0 9
1 8
2 7
3 6
4 5
9!(0,9) (0.23) (0.77) 0.0952
(9 0)!0!
9!(1,9) (0.23) (0.77) 0.2558
(9 1)!1!
9!(2,9) (0.23) (0.77) 0.3056
(9 2)!2!
9!(3,9) (0.23) (0.77) 0.2130
(9 3)!3!
9!(4,9) (0.23) (0.77) 0.0954
(9 4)!4!
( 5) 1 0
P
P
P
P
P
P x
= =−
= =−
= =−
= =−
= =−
≥ = − .0952 0.2558 0.3056 0.2130 0.0954 0.0350− − − − =
d) (9)(0.23) 2.07; (9)(0.23)(0.77) 1.26μ σ= = = =
e)
5.58
a) 2 68!(2,8) (0.62) (0.38) 0.0324
(8 2)!2!P = =
−
Discrete Probability Distributions 5-31
Copyright ©2015 Pearson Education, Inc.
b)
( )
6 2
7 1
8 0
8!(6,8) (0.62) (0.38) 0.2297
(8 6)!6!
8!(7,8) (0.62) (0.38) 0.1071
(8 7)!7!
8!(8,8) (0.62) (0.38) 0.0218
(8 8)!8!
5 0.2297 0.1071 0.0218 0.3586
P
P
P
P x
= =−
= =−
= =−
> = + + =
c)
0 8
1 7
2 6
3 5
8!(0,8) (0.62) (0.38) 0.0004
(8 0)!0!
8!(1,8) (0.62) (0.38) 0.0057
(8 1)!1!
8!(2,8) (0.62) (0.38) 0.0324
(8 2)!2!
8!(3,8) (0.62) (0.38) 0.1058
(8 3)!3!
( 3) 0.0004 0.0057 0.0324 0.1058 0.1443
P
P
P
P
P x
= =−
= =−
= =−
= =−
≤ = + + + =
d) (8)(0.62) 4.96; (8)(0.62)(0.38) 1.37μ σ= = = = e)
5-32 Chapter 5
Copyright ©2015 Pearson Education, Inc.
5.59
a) ( )36 = 10 0.9
400λ =
;
( )( )1 0.90.9 2.71828( ) (1) 0.3659
1!P x P
−
= = =
b) ( )36 = 20 1.8
400λ =
;
( )( )1 1.81.8 2.71828( ) (1) 0.2975
1!P x P
−
= = =
c) ( )36 = 50 4.5
400λ =
;
( )( )4 4.54.5 2.71828( ) (4) 0.1898
4!P x P
−
= = =
d) ( )36 = 110 9.9
400λ =
;
( )( )12 9.99.9 2.71828( ) (12) 0.0928
12!P x P
−
= = =
5.60
a) ( )( )6 11.011.0 2.71828
(6) 0.04116!
P−
= =
b)
( )( )3 5.55.5 2.71828(3) 0.1133
3!P
−
= =
c)
11.0
( 7) 1.0 ( 7) 1.0 0.0786 0.9214P x P xλ =
≥ = − < = − =
d)
11
( 10) 0.3405P xλ =
< =
5.61
a) ( )( )8 1313 2.71828
(8) 0.04578!
P−
= =
b) (0) (1) (2) (3) (4) (5) (6) (7) (8) (9) 0.1658P P P P P P P P P P+ + + + + + + + + =
c)
()
( ) 1 (0) (1) (2) (3) (4) (5) (6) (7) (8) (9)
(10) (11) (12) (13) (14) (15) 0.2364
P x P P P P P P P P P P
P P P P P P
= − + + + + + + + + + +
+ + + + + + =
d) ( )( )4 6.56.5 2.71828
(0.5)(13) 6.5; ( ) (4) 0.11184!
P x Pλ−
= = = = =
Discrete Probability Distributions 5-33
Copyright ©2015 Pearson Education, Inc.
5.62
a) ( )( )2 44 2.71828
(0.25)(16) 4; ( ) (2) 0.14652!
P x Pλ−
= = = = =
b) ( )( )4 44 2.71828
(0.25)(16) 4; ( ) (4) 0.19544!
P x Pλ−
= = = = =
c) ( )( )6 88 2.71828
(0.5)(16) 8; ( ) (6) 0.12216!
P x Pλ−
= = = = =
d) ( )( )6 1212 2.71828
(0.75)(16) 12; ( ) (6) 0.02556!
P x Pλ−
= = = = =
5.63
a) ( )( )0 3.873.87 2.71828
(0) 0.02090!
P−
= =
b)
( )( )
( )( )
( )( )
0 3.87
1 3.87
2 3.87
3.87 2.71828(0) 0.0209
0!
3.87 2.71828(1) 0.0807
1!
3.87 2.71828(2) 0.1562
2!0.0209 0.0807 0.1562 0.2578( 3)
P
P
P
P x
−
−
−
= =
= =
= =
+ + =< =
c) ( )( )0 1.9351.935 2.71828
(0) 0.14440!
P−
= =
d)
( )( )
( )( )
( )( )
0 1.935
1 1.935
2 1.935
1.935 2.71828(0) 0.1444
0!
1.935 2.71828(1) 0.2795
1!
1.935 2.71828(2) 0.2704
2!0.1444 0.2795 0.2704 0.6943( 3)
P
P
P
P x
−
−
−
= =
= =
= =
+ + =< =
5.64
a) ( )( )1 6.56.5 2.71828
(1) 0.00981!
P−
= =
5-34 Chapter 5
Copyright ©2015 Pearson Education, Inc.
b)
( )( )
( )( )
( )( )
0 6.5
1 6.5
2 6.5
6.5 2.71828(0) 0.0015
0!
6.5 2.71828(1) 0.0098
1!
6.5 2.71828(2) 0.0318
2!( 3) 0.0015 0.0098 0.0318 0.0431
P
P
P
P x
−
−
−
= =
= =
= =
< = + + =
c)
( )( )
( )( )
( )( )
( )( )
( )( )
0 6.5
1 6.5
2 6.5
3 6.5
4 6.5
( 4) 1.0 0.0015 0.0098 0.0318 0.0688 0.1118 0.7763
6.5 2.71828(0) 0.0015
0!
6.5 2.71828(1) 0.0098
1!
6.5 2.71828(2) 0.0318
2!
6.5 2.71828(3) 0.0688
3!
6.5 2.71828(4) 0.1118
4!P x
P
P
P
P
P
−
−
−
−
−
> = − − − − − =
= =
= =
= =
= =
= =
d) 6.5 2.55σ = =
5.65
a) ( )( )1 5.35.3 2.71828
(3) 0.12393!
P−
= =
Discrete Probability Distributions 5-35
Copyright ©2015 Pearson Education, Inc.
b)
( )( )
( )( )
( )( )
( )( )
0 5.3
1 5.3
2 5.3
3 5.3
5.3
5.3
5.3
5.3
2.71828(0) 0.0050
0!
2.71828(1) 0.0265
1!
2.71828(2) 0.0701
2!
2.71828(3) 0.1239
3!( 4) 0.0050 0.0265 0.0701 0.1239 0.2255
P
P
P
P
P x
−
−
−
−
= =
= =
= =
= =
< = + + + =
c)
( )( )
( )( )
( )( )
( )( )
( )( )
( )( )
0 5.3
1 5.3
2 5.3
3 5.3
4 5.3
5 5.3
5.3
5.3
5.3
5.3
5.3
5.3
2.71828(0) 0.0050
0!
2.71828(1) 0.0265
1!
2.71828(2) 0.0701
2!
2.71828(3) 0.1239
3!
2.71828(4) 0.1641
4!
2.71828(5) 0.1740
5!( 6) 1 0.0050 0.0265 0.0
P
P
P
P
P
P
P x
−
−
−
−
−
−
= =
= =
= =
= =
= =
= =
> = − − − 701 0.1239 0.1641 0.1740 0.4364− − − =
d) ( )( )1 10.610.6 2.71828
(6) 0.04916!
P−
= =
5.66
a) 1 2324!(1, 24) (0.025) (0.975) 0.3352
(24 1)!1!P = =
−
b) 4 2024!(4, 24) (0.025) (0.975) 0.0025
(24 4)!4!P = =
−
5-36 Chapter 5
Copyright ©2015 Pearson Education, Inc.
c)
( )( )( )( )1 0.60
24 0.025 0.60
0.60 2.71828(1) 0.3293
1!
np
P−
= =
= =
d)
( )( )( )( )4 0.60
24 0.025 0.60
0.60 2.71828(4) 0.0030
4!
np
P−
= =
= =
5.67
a) 0 2020!(0, 20) (0.034) (0.966) 0.5007
(20 0)!0!P = =
−
b)
( )
0 20
1 19
20!(0, 20) (0.034) (0.966) 0.5007
(20 0)!0!
20!(1, 20) (0.034) (0.966) 0.3524
(20 1)!1!
2 0.5007 0.3524 0.8531
P
P
P x
= =−
= =−
< = + =
c)
( )( )( )( )0 0.68
20 0.034 0.68
0.68 2.71828(0) 0.5066
0!
np
P−
= =
= =
d)
( )( )( )( )
( )( )
( )
0 0.68
1 0.68
20 0.034 0.68
0.68
0.68
2 0.5066 0.3445 0.8511
2.71828(0) 0.5066
0!
2.71828(1) 0.3445
1!
np
P x
P
P
−
−
= =
< = + =
= =
= =
5.68
a) 3 2124!( ) (3,24) (0.035) (0.965) 0.0411
(24 3)!3!P x P= = =
−
b) 3 (24)(0.035)((24)(0.035)) 2.71828
( ) (3) 0.04263!
P x P−
= = =
Discrete Probability Distributions 5-37
Copyright ©2015 Pearson Education, Inc.
c) The probabilities are very close. 5.69
a) 18 4 3 0 4 0
18 3
( ) (0) 0.4461C CP x P
C− −= = =
b) ( ) 18 4 3 0 4 0 18 4 3 1 4 1
18 3 18 3
( ) 1 (0) (1) 1 0.1078C C C CP x P P
C C− − − −
= − + = − + =
c) ( )( )2
18 3(3)(4) (3)(4)(18 4)0.667; 0.6763
18 18 18 1μ σ −−= =
−= =
d) HYPGEOM.DIST(0, 3, 4, 18, FALSE) = 0.4461 1 – (HYPGEOM.DIST(0, 3, 4, 18, FALSE) + HYPGEOM.DIST(1, 3, 4, 18, FALSE) ) = 1 - 0.8922 = 0.1078
5.70
a) 19 14 8 8 14 8
19 8
( ) (8) 0.0397C CP x P
C− −= = =
b) 19 14 8 6 14 6 19 14 8 7 14 7 19 14 8 8 14 8
19 8 19 8 19 8
( ) (6) (7) (8) 0.6640C C C C C CP x P P P
C C C− − − − − −= + + = + + =
c) There are only five returns that are not using the short form. If eight returns are selected, at least three returns must be short forms.
d) ( )( )2
19 8(8)(14) (8)(14)(19 14)5.895; 0.9737
19 19 19 1μ σ −−= =
−= =
e) HYPGEOM.DIST(8, 8, 14, 19, FALSE) = 0.0397 HYPGEOM.DIST(6, 8, 14, 19, FALSE) + HYPGEOM.DIST(7, 8, 14, 19, FALSE) + HYPGEOM.DIST(8, 8, 14, 19, FALSE) = 0.3973 + 0.2270 + 0.0397 = 0.6640
5.71
a) 16, 10, 6, 2;N R n x= = = = 16 10 6 2 10 2
16 6
( ) (2) 0.0843C CP x P
C− −= = =
b) 16, 10, 6, 4;N R n x= = = = 16 10 6 4 10 4
16 6
( ) (4) 0.3934C CP x P
C− −= = =
c) 16, 10, 6, 6;N R n x= = = = 16 10 6 6 10 6
16 6
( ) (6) 0.0262C CP x P
C− −= = =
d) ( )( )2
16 6(6)(10) (6)(10)(16 10)3.75; 0.968
16 16 16 1μ σ −−= =
−= =
5-38 Chapter 5
Copyright ©2015 Pearson Education, Inc.
e) HYPGEOM.DIST(2, 6, 10, 16, FALSE) = 0.0843 HYPGEOM.DIST(4, 6, 10, 16, FALSE) = 0.3934 HYPGEOM.DIST(6, 6, 10, 16, FALSE) = 0.0262
5.72
a) 22, 8, 10, 4;N R n x= = = = 22 8 10 4 8 4
22 10
( ) (4) 0.3251C CP x P
C− −= = =
b) 22, 8, 10, 6;N R n x= = = = 22 8 10 6 8 6
22 10
( ) (6) 0.0433C CP x P
C− −= = =
c) The proportion of cars (0.636) is much higher than the proportion of trucks (0.364) on the lot. Four trucks out of 10 vehicles have a higher probability than 4 cars out of 10 vehicles because the 0.40 proportion is very close to the proportion of trucks on the lot.
d) ( )( )2
22 10(10)(8) (10)(8)(22 8)3.64; 1.150
22 22 22 1μ σ −−= =
−= =
e) HYPGEOM.DIST(4, 10, 8, 22, FALSE) = 0.3251 HYPGEOM.DIST(6, 10, 8, 22, FALSE) = 0.0433
5.73
a) 53, 22, 3, 0;N R n x= = = = 53 22 3 0 22 0
53 3
( ) (0) 0.1919C CP x P
C− −= = =
b) 53, 22, 3, 3;N R n x= = = = 53 22 3 3 22 3
53 3
( ) (3) 0.0657C CP x P
C− −= = =
c) 53, 22, 3, 2;N R n x= = = = 53 22 3 2 22 2
53 3
( ) (2) 0.3057C CP x P
C− −= = =
d) ( )( )2
53 3(3)(22) (3)(22)(53 22)1.245; 0.8368
53 53 53 1μ σ −−= =
−= =
e) HYPGEOM.DIST(0, 3, 22, 53, FALSE) = 0.1919 HYPGEOM.DIST(3, 3, 22, 53, FALSE) = 0.0657 HYPGEOM.DIST(2, 3, 22, 53, FALSE) = 0.3057
5.74
a) 52, 12, 5, 0;N R n x= = = = 52 12 5 0 12 0
52 5
( ) (0) 0.2532C CP x P
C− −= = =
b) 52, 12, 5, 1;N R n x= = = = 52 12 5 1 12 1
52 5
( ) (1) 0.4220C CP x P
C− −= = =
c) 52, 12, 5, 2;N R n x= = = = 52 12 5 2 12 2
52 5
( ) (2) 0.2509C CP x P
C− −= = =
Discrete Probability Distributions 5-39
Copyright ©2015 Pearson Education, Inc.
d) 52, 12, 5, 4;N R n x= = = = 52 12 5 4 12 4
52 5
( ) (4) 0.0076C CP x P
C− −= = =
e) ( )( )2
52 5(5)(12) (5)(12)(52 12)1.154; 0.9044
52 52 52 1μ σ −−= =
−= =
f) HYPGEOM.DIST(0, 5, 12, 52, FALSE) = 0.2532 HYPGEOM.DIST(1, 5, 12, 52, FALSE) = 0.4220 HYPGEOM.DIST(2, 5, 12, 52, FALSE) = 0.2509 HYPGEOM.DIST(4, 5, 12, 52, FALSE) = 0.0076