A First Course in Probability (10th Edition)
10th Edition
ISBN: 9780134753119
Author: Sheldon Ross
Publisher: PEARSON
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Textbook Question
Chapter 1, Problem 1.7STPE
Give a combinatorial explanation of the identity
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Exercise 1
Mateo is the star player of a certain soccer team and is getting ready for a very important match after
spending several months recovering from an injury. If the player is injured again during this match, the
probability that his team wins is 0.32. If Mateo is not injured, the probability that his team loses is 0.18. In
addition, according to the team doctor, the probability that the player gets injured during the match is 0.15.
a. Draw a probability tree that properly represents the situation described above. Clearly label the
probabilities on each branch of the tree.
b. What is the probability that Mateo's team wins the match?
c. If Mateo's team loses the match, what is the probability that Mateo was injured?
d. Consider the event "Mateo is injured" and the event "Mateo's team wins the match." Are these events
independent? Clearly justify your answer.
A company devoted to the production and distribution of craft beer has decided to run a
quality-control check on a batch of 330 mL Porter beer bottles. A random sample of the
contents of 52 bottles was taken; the volume (in millilitres, mL) was measured and is shown
below:
Dato
Volumen
(ml)
Volumen
Volumen
Volumen
Dato
Dato
Dato
(ml)
(ml)
(ml)
333
14
326
27
330
40
329
2
328
15
331
28
329
41
327
3
335
16
331
29
329
42
323
4
330
17
333
30
332
43
330
5
331
18
332
31
334
44
332
6
326
19
327
32
336
45
333
7
330
20
330
33
328
46
328
8
329
21
328
34
326
47
329
9
332
22
325
35
330
48
327
10
334
23
330
36
332
49
331
11
333
24
334
37
333
50
333
12
336
25
332
38
331
51
328
13
327
26
325
39
334
52
336
Exercise 4
A company that manufactures engine parts has developed a new type of piston made from aluminium-
silicon alloys and other materials. The firm wishes to examine the mechanical properties of these pistons.
Specifically, it is interested in evaluating the yield strength of a piston while it is subjected to a constant
temperature of 250 °C. Here, the yield strength is the maximum stress, measured in megapascals (MPa), that
the piston can withstand before it deforms permanently.
From the first production batches, 15 pistons were randomly selected. Each was tested to determine its yield
strength (in MPa), giving the following results:
83.2, 90.1, 86.7, 102.4, 95.9, 91.3, 88.1, 84.6, 93.2, 92.6, 100.4, 86.5, 89.2, 96.8, 91.7.
Assuming that the yield strength of this type of piston follows a Normal distribution, construct a 96 %
confidence interval for the variance, σ², of the pistons' yield strength.
Chapter 1 Solutions
A First Course in Probability (10th Edition)
Ch. 1 - a. How many different 7-place license plates are...Ch. 1 - How many outcome sequences are possible ten a die...Ch. 1 - Twenty workers are to be assigned to 20 different...Ch. 1 - John, Jim, Jay, and Jack have formed a band...Ch. 1 - For years, telephone area codes in the United...Ch. 1 - A well-known nursery rhyme starts as follows: As I...Ch. 1 - a. In how many ways can 3 boys and 3 girls sit in...Ch. 1 - When all letters are used, how many different...Ch. 1 - A child has 12 blocks, of which 6 are black, 4 are...Ch. 1 - In how many ways can 8 people be seated in a row...
Ch. 1 - In how many ways can 3 novels. 2 mathematics...Ch. 1 - How many 3 digit numbers zyz, with x, y, z all...Ch. 1 - How many different letter permutations, of any...Ch. 1 - Five separate awards (best scholarship, best...Ch. 1 - Consider a group of 20 people. If everyone shakes...Ch. 1 - How many 5-card poker hands are there?Ch. 1 - A dance class consists of 22 students, of which 10...Ch. 1 - A student has to sell 2 books from a collection of...Ch. 1 - Seven different gifts are to be distributed among...Ch. 1 - A committee of 7, consisting of 2 Republicans, 2...Ch. 1 - From a group of 8 women and 6 men, a committee...Ch. 1 - A person has 8 friends, of whom S will be invited...Ch. 1 - Consider the grid of points shown at the top of...Ch. 1 - In Problem 23, how many different paths are there...Ch. 1 - A psychology laboratory conducting dream research...Ch. 1 - Show k=0n(nk)2k=3n Simplify k=0n(nk)xkCh. 1 - Expand (3x2+y)5.Ch. 1 - The game of bridge is played by 4 players, each of...Ch. 1 - Expand (x1+2x2+3x3)4.Ch. 1 - If 12 people are to be divided into 3 committees...Ch. 1 - If 8 new teachers are to be divided among 4...Ch. 1 - Ten weight lifters are competing in a team...Ch. 1 - Delegates from 10 countries, including Russia,...Ch. 1 - If 8 identical blackboards are to be divided among...Ch. 1 - An elevator starts at the basement with 8 people...Ch. 1 - We have 520.000 that must be invested among 4...Ch. 1 - Suppose that 10 fish are caught at a lake that...Ch. 1 - Prove the generalized version of the basic...Ch. 1 - Two experiments are to be performed. The first can...Ch. 1 - In how many ways can r objects be selected from a...Ch. 1 - There are (nr) different linear arrangements of n...Ch. 1 - Determine the number of vectors (x1,...,xn), such...Ch. 1 - How many vectors x1,...,xk are there for which...Ch. 1 - Give an analytic proof of Equation (4.1).Ch. 1 - Prove that (n+mr)=(n0)(mr)+(n1)(mr1)+...+(nr)(m0)...Ch. 1 - Use Theoretical Exercise 8 I to prove that...Ch. 1 - From a group of n people, suppose that we want to...Ch. 1 - The following identity is known as Fermats...Ch. 1 - Consider the following combinatorial identity:...Ch. 1 - Show that, for n0 ,i=0n(1)i(ni)=0 Hint: Use the...Ch. 1 - From a set of n people, a committee of size j is...Ch. 1 - Let Hn(n) be the number of vectors x1,...,xk for...Ch. 1 - Consider a tournament of n contestants in which...Ch. 1 - Present a combinatorial explanation of why...Ch. 1 - Argue...Ch. 1 - Prove the multinomial theorem.Ch. 1 - In how many ways can n identical balls be...Ch. 1 - Argue that there are exactly (rk)(n1nr+k)...Ch. 1 - Prob. 1.22TECh. 1 - Determine the number of vectors (xi,...,xn) such...Ch. 1 - How many different linear arrangements are there...Ch. 1 - If 4 Americans, 3 French people, and 3 British...Ch. 1 - A president. treasurer, and secretary. all...Ch. 1 - A student is to answer 7 out of 10 questions in an...Ch. 1 - In how many ways can a man divide 7 gifts among...Ch. 1 - How many different 7-place license plates are...Ch. 1 - Give a combinatorial explanation of the...Ch. 1 - Consider n-digit numbers where each digit is one...Ch. 1 - Consider three classes, each consisting of n...Ch. 1 - How many 5-digit numbers can be formed from the...Ch. 1 - From 10 married couples, we want to select a group...Ch. 1 - A committee of 6 people is to be chosen from a...Ch. 1 - An art collection on auction consisted of 4 Dalis,...Ch. 1 - Prob. 1.14STPECh. 1 - A total of n students are enrolled in a review...Ch. 1 - Prob. 1.16STPECh. 1 - Give an analytic verification of...Ch. 1 - In a certain community, there are 3 families...Ch. 1 - If there are no restrictions on where the digits...Ch. 1 - Verify the...Ch. 1 - Simplify n(n2)+(n3)...+(1)n+1(nn)
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