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
The weight of a random person is normally distributed , with a mean of 62 kg and variance of 15 kg2 .
What is the probability that a random person's height is over 58 kg?
Please round your result to two decimal places, and use the z-table.
Transcribed Image Text: Standard Normal Distrībution Function: (z) =Le-y*/2 dy
.00
.01
02
03
04
.05
06
07
09
.0
.5000
.5040
.5080
.5120
.5160
.5199
.5239
.5279
.5319
.5359
.1
.5398
.5438
.5478
.5517
.5557
.5596
.5636
.5675
.5714
.5753
.2
.5793
.5832
.5871
.5910
.5948
.5987
.6026
.6064
.6103
.6141
.6255
.6480
.6844
.3
.6179
.6217
.6293
.6331
.6368 .6406
.6443
.6517
4
.6554
.6591
.6628
.6664
.6700
.6736
.6772
.6808
.6879
.5
.6915
.6950
.6985
.7019
.7054
.7088
.7123
7157
.7190
.7224
.7549
.7852
.6
.7257
.7291
.7324
.7357
.7389
.7422
.7454
.7486
.7517
.7
.7580
.7611
.7642
.7673
.7704
.7734
.7764
.7794
.7823
.8
.7881
.7910
.7939
.7967
.7995
.8023
.8051
.8078
.8106
.8133
.9
.8159
.8186
.8212
.8238
.8264
.8289
.8315
.8340
.8365
8389
1.0
.8413
.8438
.8461
.8485
.8508
.8531
.8554
.8577
.8599
8621
1.1
8643
.8665
.8686
8708
.8729
.8749
.8770
8790
8810
.8830
1.2
.8849
.8869
.8888
.8907
.8925
8944
8962
.8980
.8997
.9015
1.3
9032
9049
9066
9082
9099
.9115
9131
.9147
9162
.9177
1.4
9192
9207
9222
.9236
.9251
.9265
.9279
9292
.9306
.9319
9441
9545
1.5
9332
9345
9357
9370
9382
.9394
.9406
.9418
9429
1.6
9452
9463
9474
9484
.9495
.9505
9515
9525
.9535
1.7
9554
.9564
9573
9582
9591
.9599
9608
9616
.9625
9633
1.8
.9641
9649
9656
9664
9671
9678
9686
9693
9699
.9706
1.9
9713
9719
9726
9732
.9738
9744
9750
9756
.9761
.9767
20
9772
9778
9783
9788
.9793
.9798
.9803
9808
9812
9817
21
9821
9826
9830
9834
.9838
.9842
.9846
9850
.9854
.9857
22
9861
9864
9868
9871
.9875
.9878
.9881
9884
.9887
.9890
2.3
9893
9896
9898
9901
.9904
.9906
.9909
.9911
.9913
.9916
24
9918
9920
9922
9925
.9927
9929
.9931
9932
.9934
9936
25
.9938
9940
9941
9943
.9945
.9946
.9948
.9949
.9951
.9952
26
.9953
.9955
9956
9957
9960
.9963
9959
9969
.9961
.9962
.9964
27
9965
9966
.9967
9968
.9970
.9971
.9972
.9973
.9974
2.8
.9974
9975
.9976
.9977
9977
.9978
.9979
.9979
.9980
9981
29
9981
9982
9982
9983
.9984
9984
.9985
9985
9986
.9986
3.0
.9987
9987
9987
.9988
.9988
.9989
.9989
.9989
.9990
.9990
3.1
9990
9991
9991
9991
.9992
.9992
.9992
.9992
.9993
.9993
9993
9995
3.2
9993
9994
.9994
9994
.9994
9994
9995
9995
.9995
3.3
.9995
.9995
.9996
9996
.9996
.9996
.9996
9996
.9997
3.4
.9997
9997
9997
.9997
.9997
.9997
.9997
.9997
.9997
.9998
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.
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