1 Combinatorial Analysis 2 Axioms Of Probability 3 Conditional Probability And Independence 4 Random Variables 5 Continuous Random Variables 6 Jointly Distributed Random Variables 7 Properties Of Expectation 8 Limit Theorems 9 Additional Topics In Probability 10 Simulation Chapter1: Combinatorial Analysis
Chapter Questions Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and... Problem 1.2P: How many outcome sequences are possible ten a die is rolled four times, where we say, for instance,... Problem 1.3P: Twenty workers are to be assigned to 20 different jobs, one to each job. How many different... Problem 1.4P: John, Jim, Jay, and Jack have formed a band consisting of 4 instruments if each of the boys can play... Problem 1.5P: For years, telephone area codes in the United States and Canada consisted of a sequence of three... Problem 1.6P: A well-known nursery rhyme starts as follows: As I was going to St. Ives I met a man with 7 wives.... Problem 1.7P: a. In how many ways can 3 boys and 3 girls sit in a row? b. In how many ways can 3 boys and 3 girls... Problem 1.8P: When all letters are used, how many different letter arrangements can be made from the letters a.... Problem 1.9P: A child has 12 blocks, of which 6 are black, 4 are red, 1 is white, and 1 is blue. If the child puts... Problem 1.10P: In how many ways can 8 people be seated in a row if a. there are no restrictions on the seating... Problem 1.11P: In how many ways can 3 novels. 2 mathematics books, and 1 chemistry book be arranged on a bookshelf... Problem 1.12P: How many 3 digit numbers zyz, with x, y, z all ranging from 0 to9 have at least 2 of their digits... Problem 1.13P: How many different letter permutations, of any length, can be made using the letters M 0 T T 0. (For... Problem 1.14P: Five separate awards (best scholarship, best leadership qualities, and so on) are to be presented to... Problem 1.15P: Consider a group of 20 people. If everyone shakes hands with everyone else, how many handshakes take... Problem 1.16P: How many 5-card poker hands are there? Problem 1.17P: A dance class consists of 22 students, of which 10 are women and 12 are men. If 5 men and 5 women... Problem 1.18P: A student has to sell 2 books from a collection of 6 math, 7 science, and 4 economics books. How... Problem 1.19P: Seven different gifts are to be distributed among 10 children. How many distinct results are... Problem 1.20P: A committee of 7, consisting of 2 Republicans, 2 Democrats, and 3 Independents, is to be chosen from... Problem 1.21P: From a group of 8 women and 6 men, a committee consisting of 3 men and 3 women is to be formed. How... Problem 1.22P: A person has 8 friends, of whom S will be invited to a party. a. How many choices are there if 2 of... Problem 1.23P: Consider the grid of points shown at the top of the next column. Suppose that, starting at the point... Problem 1.24P: In Problem 23, how many different paths are there from A to B that go through the point circled in... Problem 1.25P: A psychology laboratory conducting dream research contains 3 rooms, with 2 beds in each room. If 3... Problem 1.26P: Show k=0n(nk)2k=3n Simplify k=0n(nk)xk Problem 1.27P: Expand (3x2+y)5. Problem 1.28P: The game of bridge is played by 4 players, each of w1om is dealt 13 cards. How many bridge deals are... Problem 1.29P: Expand (x1+2x2+3x3)4. Problem 1.30P: If 12 people are to be divided into 3 committees of respective sizes 3, 4, and 5, how many divisions... Problem 1.31P: If 8 new teachers are to be divided among 4 schools, how many divisions are possible? What if each... Problem 1.32P: Ten weight lifters are competing in a team weight-lifting contest. Of the lifters, 3 are from the... Problem 1.33P: Delegates from 10 countries, including Russia, France, England, and the United States, are to be... Problem 1.34P: If 8 identical blackboards are to be divided among 4 schools, how many divisions are possible? How... Problem 1.35P: An elevator starts at the basement with 8 people (not including the elevator operator) and... Problem 1.36P: We have 520.000 that must be invested among 4 possible opportunities. Each investment must be... Problem 1.37P: Suppose that 10 fish are caught at a lake that contains 5 distinct types of fish. a. How many... Problem 1.1TE: Prove the generalized version of the basic counting principle. Problem 1.2TE: Two experiments are to be performed. The first can result in any one of m possible outcomes. If the... Problem 1.3TE: In how many ways can r objects be selected from a set of n objects if the order of selection is... Problem 1.4TE: There are (nr) different linear arrangements of n balls of which r are black and nr are white. Give... Problem 1.5TE: Determine the number of vectors (x1,...,xn), such that each x1 is either 0 or 1 andi=1nxiK Problem 1.6TE: How many vectors x1,...,xk are there for which each xi is a positive integer such that1xin and... Problem 1.7TE: Give an analytic proof of Equation (4.1). Problem 1.8TE: Prove that (n+mr)=(n0)(mr)+(n1)(mr1)+...+(nr)(m0) Hint: Consider a group of n men and m women. How... Problem 1.9TE: Use Theoretical Exercise 8 I to prove that (2nn)=k=0n(nk)2 Problem 1.10TE: From a group of n people, suppose that we want to choose a committee of k,kn, one of whom is to be... Problem 1.11TE: The following identity is known as Fermats combinatorial identity:(nk)=i=kn(i1k1)nk Give a... Problem 1.12TE: Consider the following combinatorial identity: k=0nk(nk)=n2n1 a. Present a combinatorial argument... Problem 1.13TE: Show that, for n0 ,i=0n(1)i(ni)=0 Hint: Use the binomial theorem. Problem 1.14TE: From a set of n people, a committee of size j is to be chosen, and from this committee, a... Problem 1.15TE: Let Hn(n) be the number of vectors x1,...,xk for which each xi is a positive integer satisfying 1xin... Problem 1.16TE: Consider a tournament of n contestants in which the outcome is an ordering of these contestants,... Problem 1.17TE: Present a combinatorial explanation of why (nr)=(nr,nr) Problem 1.18TE: Argue that(nn1,n2,...,nr)=(n1n11,n2,...,nr)+(nn1,n21,...,nr)+...+(nn1,n2,...,nr1) Hint: Use an... Problem 1.19TE: Prove the multinomial theorem. Problem 1.20TE: In how many ways can n identical balls be distributed into r urns so that the ith urn contains at... Problem 1.21TE: Argue that there are exactly (rk)(n1nr+k) solutions of x1+x2+...+xr=n for which exactly k of the xi... Problem 1.22TE Problem 1.23TE: Determine the number of vectors (xi,...,xn) such that each xi, is a nonnegative integer and i=1nxik. Problem 1.1STPE: How many different linear arrangements are there of the letters A, B, C, D, E, F for which a. A and... Problem 1.2STPE: If 4 Americans, 3 French people, and 3 British people are to be seated in a row, how many seating... Problem 1.3STPE: A president. treasurer, and secretary. all different, are to be chosen from a club onsisting of 10... Problem 1.4STPE: A student is to answer 7 out of 10 questions in an examination. How many choices has she? How many... Problem 1.5STPE: In how many ways can a man divide 7 gifts among his 3 children if the eldest is to receive 3 gifts... Problem 1.6STPE: How many different 7-place license plates are possible mien 3 of the entries are letters and 4 are... Problem 1.7STPE: Give a combinatorial explanation of the identity(nr)=(nnr) Problem 1.8STPE: Consider n-digit numbers where each digit is one of the 10 integers 0,1, ... ,9. How many such... Problem 1.9STPE: Consider three classes, each consisting of n students. From this group of 3n students, a group of 3... Problem 1.10STPE: How many 5-digit numbers can be formed from the integers 1,2,... ,9 if no digit can appear more than... Problem 1.11STPE: From 10 married couples, we want to select a group of 6 people that is not allowed to contain a... Problem 1.12STPE: A committee of 6 people is to be chosen from a group consisting of 7 men and 8 women. If the... Problem 1.13STPE: An art collection on auction consisted of 4 Dalis, 5 van Goghs. and 6 Picassos, At the auction were... Problem 1.14STPE Problem 1.15STPE: A total of n students are enrolled in a review course for the actuarial examination in probability.... Problem 1.16STPE Problem 1.17STPE: Give an analytic verification of (n2)=(k2)+k(nk)+(n+k2),1kn. Now, give a combinatorial argument for... Problem 1.18STPE: In a certain community, there are 3 families consisting of a single parent and 1 child, 3 families... Problem 1.19STPE: If there are no restrictions on where the digits and letters are placed, how many 8-place license... Problem 1.20STPE: Verify the identityx1+...+xr=n,xi0n!x1!x2!...xr!=rn a. by a combinatorial argument that first notes... Problem 1.21STPE: Simplify n(n2)+(n3)...+(1)n+1(nn) Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
Related questions
A simple random sample from a population with a normal distribution of 97 body temperatures has x=98.50°F and s=0.61°F. Construct a 98% confidence interval estimate of the standard deviation of body temperature of all healthy humans.
Transcribed Image Text: 34.
A simple random sample from a population with a normal distribution of 97 body temperatures has x = 98.50°F and s= 0.61°F. Construct a 98% confidence interval estimate of
the standard deviation of body temperature of all healthy humans.
12 Click the icon to view the table of Chi-Square critical values.
°F <o<
°F
(Round to two decimal places as needed.)
Transcribed Image Text: 6
Degrees of
Freedom
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
67
17
18
19
20
21
22
23
24
25
26
27
28
29
30
40
50
60
70
80
90
100
Degrees of
Freedom
Chi-Square (X2) Distribution
0.995
0.010
0.072
0.207
0.412
0.676
0.989
1.344
1.735
2.156
2.603
3.074
3.565
4.075
4.601
5.142
5.697
6.265
6.844
7.434
8.034
8.643
9.260
9.886
10.520
11.160
11.808
12.461
13.121
13.787
20.707
27.991
35.534
43.275
51.172
59.196
67.328
0.995
(r- 1)(c-1)
k-1
Degrees of Freedom
n-1
k-1
0.99
0.020
0.115
0.297
0.554
0.872
1.239
1.646
2.088
2.558
3.053
3.571
4.107
4.660
5.229
5.812
6.408
7.015
7.633
8.260
8.897
9.542
10.196
10.856
11.524
12.198
12.879
13.565
14.257
14.954
22.164
29.707
37.485
45.442
53.540
61.754
70.065
0.99
0.975
0.001
0.051
0.216
0.484
0.831
1.237
1.690
2.180
2.700
3.247
3.816
4.404
5.009
5.629
6.262
6.908
7.564
8.231
8.907
9.591
10.283
10.982
11.689
12.401
13.120
13.844
14.573
15.308
16.047
16.791
24.433
32.357
40.482
48.758
57.153
65.647
74.222
0.975
Area to the Right of the Critical Value
0.95
0.004
0.103
0.352
0.711
1.145
1.635
2.167
2.733
3.325
3.940
4.575
5.226
5.892
6.571
7.261
7.962
8.672
9.390
10.117
10.851
11.591
12.338
13.091
13.848
14.611
15.379
16.151
16.928
17.708
18.493
26.509
34.764
43.188
51.739
60.391
69.126
77.929
0.90
0.016
0.211
0.584
1.064
1.610
2.204
2.833
3.490
4.168
4.865
5.578
Contingency table test with rrows and c columns
Kruskal-Wallis test with k different samples
6.304
7.042
7.790
8.547
9.312
10.085
10.865
11.651
12.443
13.240
14.042
14.848
15.659
16.473
17.292
18.114
18.939
19.768
20.599
29.051
37.689
46.459
55.329
64.278
73.291
82.358
Confidence Interval or Hypothesis Test for a standard deviation or variance
Goodness-of-fit test with different categories
0.10
2.706
4.605
6.251
7.779
9.236
10.645
12.017
13.362
14.684
15.987
17.275
18.549
19.812
21.064
22.307
23.542
24.769
25.989
27.204
28.412
29.615
30.813
32.007
33.196
34.382
35.563
36.741
37.916
39.087
40.256
0.95
0.90
Area to the Right of the Critical Value
51.805
63.167
74.397
85.527
96.578
107.565
118.498
0.10
0.05
3.841
5.991
7.815
9.488
11.071
12.592
14.067
15.507
16.919
18.307
19.675
21.026
22.362
23.685
24.996
26.296
27.587
28.869
30.144
31.410
32.671
33.924
35.172
36.415
37.652
38.885
40.113
41.337
42.557
43.773
55.758
67.505
79.082
90.531
101.879
113.145
124.342
0.05
0.025
5.024
7.378
9.348
11.143
12.833
14.449
16.013
17.535
19.023
20.483
21.920
23.337
24.736
26.119
27.488
28.845
30.191
31.526
32.852
34.170
35.479
36.781
38.076
39.364
40.646
41.923
43.194
44.461
45.722
46.979
59.342
71.420
83.298
95.023
106.629
118.136
129.561
0.025
0.01
6.635
9.210
11.345
13.277
15.086
16.812
18.475
20.090
21.666
23.209
24.725
26.217
27.688
29.141
30.578
32.000
33.409
34.805
36.191
37.566
38.932
40.289
41.638
42.980
44.314
45.642
46.963
48.278
49.588
50.892
63.691
76.154
88.379
100.425
112.329
124.116
135.807
0.01
0.005
7.879
10.597
12.838
14.860
16.750
18.548
20.278
21.955
23.589
25.188
26.757
28.299
29.819
31.319
32.801
34.267
35.718
37.156
38.582
39.997
41.401
42.796
44.181
45.559
46.928
48.290
49.645
50.993
52.336
53.672
66.766
79.490
91.952
104.215
116.321
128.299
140.169
0.005
Degrees of
Freedom
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
1 2 2 2 2 2 2 2 2 2 2 8 2888
17
18
19
20
21
22
23
24
25
26
27
28
29
30
40
50
60
70
80
90
100
Degrees of
Freedom
Features Features Normal distribution is characterized by two parameters, mean (µ) and standard deviation (σ). When graphed, the mean represents the center of the bell curve and the graph is perfectly symmetric about the center. The mean, median, and mode are all equal for a normal distribution. The standard deviation measures the data's spread from the center. The higher the standard deviation, the more the data is spread out and the flatter the bell curve looks. Variance is another commonly used measure of the spread of the distribution and is equal to the square of the standard deviation.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 2 images
