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
1a. Fish weights have a mean of 3 pounds and a standard deviation of 0.5 pound. What percentage of these fish should be heavier than 4 pounds?
Selection Group: Is the correct choice = 30.85%; 2.28%; 2.00%; 97.73%; or 69.15%
1b. Heights at the shoulder for adults of a breed of dog are normally distributed with a mean of 22 inches and standard deviation of 4 inches. What shoulder height is the boundary of the tallest 10% of adult dogs of this breed? (be as close as our chart allows)
Group of answer choices: Is the correct answer = 31.32 in; 26.00 in; 25.60 in; 27.12 in; or 16.88 in
Transcribed Image Text: Cumulative Standard Normal Distribution Table
s
0.01
0.02
0.03
0.04 0.05
0.06
0.07
0.09
0.08
0.4721 0.4681 0.4641
0.4325 0.4286
0.4247
0.3974 0.3936 0.3897
0.3859
0.3483
-0.50
0.0367
0.0294
Z
0.00
-0.00 0.5000 0.4960 0.4920 0.4880 0.4840
0.4801 0.4761
-0.10 0.4602 0.4562 0.4522 0.4483 0.4443 0.4404 0.4364
-0.20
0.4207 0.4168 0.4129 0.4090 0.4052 0.4013
-0.30 0.3821 0.3783 0.3745 0.3707 0.3669 0.3632 0.3594 0.3557 0.3520
-0.40 0.3446 0.3409 0.3372 0.3336 0.3300 0.3264 0.3228 0.3192 0.3156 0.3121
0.3085 0.3050 0.3015 0.2981 0.2946 0.2912 0.2877 0.2843 0.2810 0.2776
-0.60 0.2743 0.2709 0.2676 0.2643 0.2611 0.2578 0.2546 0.2514 0.2483 0.2451
-0.70 0.2420 0.2389 0.2358 0.2327 0.2296 0.2266 0.2236 0.2206 0.2177 0.2148
-0.80 0.2119 0.2090 0.2061 0.2033 0.2005 0.1977 0.1949 0.1922 0.1894 0.1867
-0.90 0.1841 0.1814 0.1788 0.1762 0.1736 0.1711 0.1685 0.1660 0.1635 0.1611
-1.00 0.1587 0.1562 0.1539 0.1515 0.1492 0.1469 0.1446 0.1423 0.1401
0.1379
-1.10 0.1357 0.1335 0.1314 0.1292 0.1271 0.1251 0.1230 0.1210 0.1190 0.1170
-1.20 0.1151 0.1131 0.1112 0.1093 0.1075 0.1056 0.1038 0.1020 0.1003 0.0985
-1.30 0.0968 0.0951 0.0934 0.0918 0.0901 0.0885 0.0869 0.0853 0.0838 0.0823
-1.40 0.0808 0.0793 0.0778 0.0764 0.0749 0.0735 0.0721 0.0708 0.0694 0.0681
-1.50 0.0668 0.0655 0.0643 0.0630 0.0618 0.0606 0.0594 0.0582 0.0571 0.0559
-1.60 0.0548 0.0537 0.0526 0.0516 0.0505 0.0495 0.0485 0.0475 0.0465 0.0455
-1.70 0.0446 0.0436 0.0427 0.0418 0.0409 0.0401 0.0392 0.0384 0.0375
-1.80 0.0359 0.0351 0.0344 0.0336 0.0329 0.0322 0.0314 0.0307 0.0301
-1.90 0.0287 0.0281 0.0274 0.0268 0.0262 0.0256 0.0250 0.0244 0.0239
-2.00 0.0228 0.0222 0.0217 0.0212 0.0207 0.0202 0.0197 0.0192 0.0188 0.0183
-2.10 0.0179 0.0174 0.0170 0.0166 0.0162 0.0158 0.0154 0.0150 0.0146 0.0143
-2.20 0.0139 0.0136 0.0132 0.0129 0.0125 0.0122 0.0119 0.0116 0.0113 0.0110
-2.30 0.0107 0.0104 0.0102 0.0099 0.0096 0.0094 0.0091 0.0089 0.0087 0.0084
-2.40 0.0082 0.0080 0.0078 0.0075 0.0073 0.0071 0.0069 0.0068 0.0066 0.0064
-2.50
0.0062 0.0060 0.0059 0.0057 0.0055 0.0054 0.0052 0.0051 0.0049 0.0048
-2.60 0.0047 0.0045 0.0044 0.0043 0.0041 0.0040 0.0039 0.0038 0.0037 0.0036
-2.70 0.0035 0.0034 0.0033 0.0032 0.0031 0.0030 0.0029 0.0028 0.0027 0.0026
-2.80 0.0026 0.0025 0.0024 0.0023 0.0023 0.0022 0.0021 0.0021 0.0020 0.0019
-2.90 0.0019 0.0018 0.0018 0.0017 0.0016 0.0016 0.0015 0.0015 0.0014 0.0014
-3.00 0.0013 0.0013 0.0013 0.0012 0.0012 0.0011 0.0011 0.0011 0.0010 0.0010
-3.10 0.0010 0.0009 0.0009 0.0009 0.0008 0.0008 0.0008 0.0008 0.0007 0.0007
-3.20 0.0007 0.0007 0.0006 0.0006 0.0006 0.0006 0.0006 0.0005 0.0005 0.0005
-3.30 0.0005 0.0005 0.0005 0.0004 0.0004 0.0004 0.0004 0.0004 0.0004 0.0003
-3.40 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0002
-3.50
0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002
-3.60
0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001
-3.70 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001
-3.80 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001
*Note: z-values less than -3.89 produce a probability of zero.
0.0233
0.0002 0.0002
0.0002 0.0002
Department of Mathematics, Sinclair Community College, Dayton, OH
Transcribed Image Text: TABLE 1 CUMULATIVE PROBABILITIES FOR THE STANDARD NORMAL
DISTRIBUTION (Continued)
Z
4789 BAA93 29539 87 388 3
5
6
1.0
1.1
1.2
.0
.1
5000 5040 .5080 5120 5160
5398 5438 5478 .5517 5557
5793 5832 5871 5910 5948
.3 .6179 .6217 .6255 .6293 .6331
4 .6554 .6591
.2
.6628 .6664 .6700
1.3
.00
1.6
1.8
.01
2.5 9938
2.6 .9953
2.7 .9965
2.8 .9974
2.9 .9981
.02
3.0 9987
0
.03
Z
Cumulative
probability
.04
.7157 .7190
.6915 6950
7257 .7291
.7088 .7123
.7422 .7454
7486 7517
.8
.8621
9177
9319
6985 .7019 .7054
7324 .7357 .7389
7 .7580 7611 .7642 .7673 .7704 .7734 .7764 .7794 .7823
7881 .7910 .7939 .7967 .7995 8023 .8051 .8078 .8106
.8159 .8186 .8212 .8238 .8264 .8289 .8315 .8340 .8365
8413 .8438 .8461
.8485
.8508 .8531 .8554 .8577 .8599
.8643 .8665 .8686 .8708 .8729 .8749 8770 .8790 .8810 .8830
.8849 .8869 .8888 .8907 .8925 .8944 .8962 .8980 .8997 9015
.9032 9049 .9066 9082 .9099 9115 .9131 9147 9162
1.4 .9192 .9207 .9222 .9236 .9251 .9265 .9279 .9292 9306
1.5 .9332 9345 9357 9370 9382 9394 .9406 9418 9429 9441
9452 .9463
9474 9484 9495 .9505 9515 .9525 .9535 .9545
1.7 .9554 .9564 .9573 9582 .9591 .9599 .9608 .9616 9625 9633
.9641 .9649 .9656 9664 .9671 .9678 .9686 .9693 .9699 .9706
1.9 .9713 .9719 .9726 .9732 9738 .9744 9750 9756 9761 .9767
2.0 9772 9778 9783 9788 9793
2.1 9821 .9826 .9830 .9834 .9838
2.2 9861 9864 9868 9871 9875
2.3 9893 .9896 9898 9901 9904
2.4 9918 9920 9922 9925 9927
.9940 9941 .9943 9945 .9946 9948
.9955 9956 9957 9959 .9960 .9961
.9966 9967 .9968 9969 .9970 9971 .9972 .9973
.9975 .9976 .9977 9977 .9978 .9979 9979 9980
.9982 .9982 .9983 .9984 .9984 9985 9985 .9986
9987 .9987 .9988
.9949 9951
.9962 9963
.9988 9989
9989 9989
.9990 .9990
.05
Entries in the table
give the area under the
curve to the left of the
z value. For example, for
z = 1.25, the cumulative
probability is .8944.
.06
.07
.08
.5199 5239 .5279 5319
.5596 5636 .5675 5714
.5987 .6026 .6064 .6103
.6368
.6736
.6406 .6443
.6808
.6772
.09
5359
.5753
.6141
.6480 6517
.6844 .6879
.7224
.7549
7852
.8133
.8389
9817
.9798 .9803 .9808 9812
.9842 .9846 .9850 .9854 .9857
9878 .9881 .9884 9887 .9890
.9906 .9909 .9911 9913 9916
.9929 .9931 .9932 9934 9936
9952
9964
9974
.9981
.9986
Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...
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