All Matches

Solution Library

Expert Answer

Textbooks

Search Textbook questions, tutors and Books

Oops, something went wrong!

Change your search query and then try again

Result Not Found

Books FREE
Tutors
Study Help
Hire a Tutor
AI Study Help New

Search
Sign In
Register

study help
business
introduction to probability statistics

Questions and Answers of Introduction To Probability Statistics

Among 15 couples who attend a party, we choose randomly 4 persons. What is the probability that among them(i) there are only men?(ii) there are only women?(iii) there is at least one couple?(Remember
A drawer contains six pairs of gloves, each of a different color. A person selects randomly 4 gloves among the 12. By recording all possible and all favorable events in each case (as in the previous
The creation and enumeration of all possible combinations from a set can be done in Mathematica with the command KSubsets. More specifically, the command KSubsets[A, k] gives all subsets of a set A
Generalizing the Chevalier de Méré problem (see Exercise 16 in Section 2.3), suppose pk denotes the probability that we get at least one six when we throw a die k times, and qk denotes the
Recall the birthday problem (see Example 2.12). Let 1 − pk denote the probability that in a group of k students, at least two share the same birthday. This probability is found below using
Working as in the previous exercise, answer the following questions:(i) An urn contains 10 balls numbered 0–9. We select a ball at random, note the number on it, and return the ball into the urn.
The next program finds all possible outcomes in the experiment of throwing two dice. Then, it calculates the number of outcomes in which the sum of the dice equals six as well as the probability of
Counting of all possible permutations of a set of elements (with or without repetitions) can be done with the following sequence of commands:In[1]:= perm=Permutations[{a,b,c}];Print["The permutations
The following sequence of commands Do[ a=Binomial[Binomial[n,2],2]; b=Binomial[(n+1),4];Print[n," ",a," ",b," ",a-3*b],{n,3,10}]can be used to verify numerically the validity of the identity(see Part
It is apparent that, as n grows large, the factorial n! can be astronomical. For instance, 20! is about 2.43 × 1018, while 100! is around 9.3 × 10157. Mathematica can be used to find the binomial
Let r and n be two nonnegative integers and p and q be two positive real numbers such that p + q = 1. Then, show that (k), (" + k 1 ) p qrk = k=r k = n(n+1)(n+2)(n+ k 1)p".
Let r and n be two nonnegative integers such that r ≤ n and x ∈ ℝ. Using (2.15), calculate the following sums:Use these results and those of the previous exercise to calculate each of the
Using the identity in (2.15), calculate the following sums: and n n (2) -1) (1) k=0 n k=0 k(k-1)(k-2) ("). k=0
Assume that n is a positive integer. Then, use the Cauchy formula to calculate the following sums: (i) (ii) n - ()(); k=1 (;). k=0 k -
By choosing appropriate values fora, b, and t in the binomial expansions of Propositions 2.9 and 2.10, calculate the following sums (n here is a nonnegative integer): n () 3*-*; k=0 n+. - (ii) (+1)
The formula by A.T. Vandermonde (1737–1796) expresses the number of permutations of m + n elements in terms of the numbers of permutations of m elements and that of n elements, as follows:Show that
In the binomial expansion of the following quantities, identify the coefficient of the constant term, i.e. the term which does not involve x: 15 i) (21 - 44) '5; (ii) (4x-3x-2)15.
How many different subsets does a set of n elements (such as the set {1, 2,…, n})have?
Suppose that each side of every piece in the domino game (see Example 2.16)is marked with a number of spots, where this number is chosen from the set{0, 1, 2,…, n − 1}. For which value of n, 66
In the warehouse of a car tyre factory, there are 1200 tyres of a certain type. Among these, there are 25 tyres that have a defect, 30 tyres for which the sell-by date has expired, while the
A committee of 5 people is to be chosen from a group of 12 people. How many different selections are available if among the 12 persons there are two who do not agree to serve in the committee
Twenty couples attend a Halloween party and enter a competition for the best outfit of the night. There are three prizes for the best outfits and prizes are won individually. What is the probability
In a lottery that has 49 numbers, we select six numbers at random and buy a ticket.What is the probability that we get(i) six winning numbers?(ii) five winning numbers?(iii) four winning numbers?(iv)
For positive integers n and k, prove the following identities: (i) (ii) (iii) k (iv) (v) (vi) n k+1 A-3 k n n k n n n k [23] - n k n- k (n+k-1) n+ - n+k-1 n-1 (2n)! (n! 2) n n 2; - n D[k1]; n k -
An usual pack of cards has 26 black cards and 26 red ones. If we split the pack into two halves with 26 each, what is the probability that each of these parts has exactly 13 black cards?
From an usual pack of 52 cards, we select 5 cards. What is the probability that at least one King is selected if(i) after each card is chosen, the card is put back for the next selection?(ii) when a
A company has 25 trucks, among which 5 have fuel emissions above a certain level. If one of the company’s technicians selects six trucks at random to check for their fuel emissions, what is the
Mary has 10 coins in her pocket, of which 4 are gold ones and 6 are silver ones.She selects four coins at random. Find the probability that exactly two of them are gold if(i) the selection of the
In a forested area, it is known that there exist 300 animals from a protected species.A scientific team selects 100 of these animals, marks them and sets them free. After a certain period, so that
In the Mathematics Department of a University, there are 80 freshmen, 65 sophomore, 70 junior, and 90 senior students. If five students have been chosen for the chess team, what is the probability
(i) For any positive integer n, show that(ii) For n ≥ 6, verify that the following inequality holds: ((2)` 222 = 3 ("+). 4
For nonnegative integers n, k, and r, prove the following identities: 1k (^) = n (n = 1), 1 k n; *(*)="(*) n-k+1 (ii) (")=(1), 1kn; (iii) k n- (*)=("), 0 k < n; n-k (iv) (*) (*) = (*) (*), 0r k n;
If n and k are positive integers, reduce the following sums to an expression involving just one combinatorial term: and ('12")+(12)+(112) n (1)+()+(-1). k
A box has 25 balls numbered 1–25. Suppose four balls are selected randomly without replacement. What is the probability that the smallest number in the balls chosen is at least six?
A company has 12 senior and 20 junior employees. The members of staff in the company want to form a five-member committee. Find how many different ways are possible to select the committee, under the
At theNational Basketball Association (NBA) championships, 30 teams participate, 15 of which belong to the Eastern Conference and 15 to the Western Conference.At the end of the regular season, the
At the repair department of a store that sells electric appliances, there are currently six TV sets and eight DVD players waiting to be repaired. The staff of the store can repair a total of six
In a volleyball tournament, six teams take part and each team has to play every other team exactly once. How many matches will be played in total?
In the birthday problem (Example 2.12), use the approximation ex ≅ 1 + x (valid when x is small) to obtain an approximation for the probability pk in (2.5) as pk ≅ e−k(k−1)∕(2×365).Check
In relation to the second event of the previous problem, suppose now that the number of throws is not fixed and consider more generally the event Ak: we get at least one double six when we throw two
(The Chevalier de Méré problem4) Which of the following two events is more likely to occur?(i) we get at least one six when we throw a die four times;(ii) we get at least one double six when we
Suppose that n and k are two positive integers with k ≤ n. Verify the truth of the following identities:(i) (n + 1)k = (n)k + k(n)k−1;(ii) (n + 1)k = k! + k[(n)k−1 + (n − 1)k−1 + · · · +
Find the integer k that satisfies the equality (*) 16 3 (8)
Suppose that Carol belongs to a class of k students. What is the probability that at least one of Carol’s classmates has his/her birthday the same day as Carol?Compare the result with that found in
Ahigh-school class is attended by 10 boys B1, B2,…, B10 and 5 girlsG1,G2,…,G5.After their final exams, all students are graded and ranked from 1 to 15.(i) How many different rankings are
We have k envelopes numbered 1 to k and n typed letters that are numbered 1 to n.In each envelope, we may put from 0 up to n letters.(i) Under the above assumptions, in how many different ways the
A bowl contains nine stones, numbered 1–9. We select successively six stones, without replacement, and write down the six-digit number that is produced from the numbers on the selected stones (in
A bus starts its route with k persons. The number of stops in this route, including the final one where all remaining passengers get off, is n.(i) Calculate the number of different ways the k persons
A father of five children has bought seven gifts. How many distinct gift allocations are possible if each child is to receive exactly one gift (so that 2 gifts will not be given to anyone)?
In a company of k students with k ≤ 12, what is the probability that at least two of them have their birthday in the same month of the year? (Assume that all months are equally likely for a
For any nonnegative integer, n, show that (3n)! is divisible by 2n3n.
Let k and n be positive integers such that 1 ≤ k ≤ n. Verify that the following relations are true:(i) (n + 1)! = (n + 1)n!;(ii) (n)1 = n;(iii) (n)n−1 = n!;(iv) (n)k = n ⋅ (n − 1)k−1, k
We ask a person to select a nonnegative integer less than 1 000 000. If all numbers have the same probability to be chosen, find the probability that the number selected does not contain the digit 5.
What is the probability that three randomly chosen persons had their last birthday on different days of the week?
We throw a die three times. What is the probability that(i) a six does not appear in any of the three throws?(ii) six appears at least once in the three throws?
A mother of three young children buys three presents for them for Christmas. She then asks her children to write down, in a piece of paper, which of the three presents they prefer, so that each one
Consider now the sample spaceΩ = {(x, y, z) ∶ x, y, z ∈ {1, 2,…, k}},where k is a positive integer. We define the events Ai = {(i, y, z) ∈ Ω ∶ y (i) Arguing as in part (i) of the previous
Consider the sample spacewhere k is a positive integer. We define the events Ai = {(i, y) ∈ Ω ∶ y (i) Using either the multiplicative law or the result of Proposition 2.3, find the number of
Suppose that we wish to make a three-letter word in the following way. The first letter is chosen among the first nine letters of the Latin alphabet (A, B, …, H, I), the second letter among the
Maria wants to select a number that has four digits and all these digits belong to the set {1, 2, 3, 4, 5, 6}. Assuming that all selections are equally likely, what is the probability that the number
Nick throws a die four times in succession.(i) What is the probability that the first three outcomes are 3, 2, and 5 (in that order)?(ii) What is the probability that at least two of the four
A psychologist is carrying out a memory test using three-letter “words” (either meaningful or not). For the first letter, she chooses among the letters G, H, D, L, the second letter can be one of
A telecommunications channel transmits digits that are either 0 or 1. A sequence of four digits is transmitted.(i) Identify the sample space Ω for this experiment.(ii) Assuming that the elements of
In a certain country, vehicle registration plates consist of three letters and four digits(e.g. ABC1234). What is the probability that a randomly chosen license plate(i) starts with A, E, or I?(ii)
A car manufacturer offers three versions for a certain type of car: normal (N), luxury(L), and executive (E). Each of the three versions of this type may be equipped with any of the four engine
Peter and Wendy have applied for a job in two different posts that have been advertised; including them, for the first post there are 12 applicants, while for the second there are 9 applicants.
A University exam consists of n multiple choice questions. For the first question there are k1 possible answers, and for the second one there are k2 answers and so on, and finally for the nth
For the council of the International Mathematical Society, there are 6 candidates for President, among whom 2 are women, 5 nominations for Vice President, 3 of whom are women, and 10 candidates are
We consider the random experiment of selecting a real number from the intervalΩ = (0, 1). Such a selection corresponds to a nonterminating process during which we successively select the first,
An urn contains 2000 lottery tickets, numbered from 1 to 2000. If we select a ticket at random, find the probability that the number on the ticket(i) is a multiple of 6;(ii) ends in 3 or 7;(iii) does
A building has two elevators: one for the first two floors only, and the other for floors three to five. Two persons enter the building and they both take the second elevator.(i) Write down all nine
In the experiment of throwing a die twice,(i) write down all 36 possible outcomes that may appear;(ii) assuming that these outcomes are equiprobable, calculate the probability that(a) at least one
Consider the situation in Example 1.6 with the emission of digital signals. In this case, calculate the probabilities of the events Ai, for i = 0, 1, 2, 3, 4, as well as those of the events B,C,D,
When tossing a coin three times, find the probability of each elementary event and then calculate the probability that(i) all three outcomes are the same (i.e. either three heads or three tails);(ii)
The cholesterol level of a person, after a blood test, can be classified as normal, acceptable, or high. From the data of previous blood tests in a hospital, it has been estimated that the
The sample space of an experiment is Ω = {5, 6, 7,…}. If the probability that the event {i} occurs is three times the probability that the event {i + 1} occurs, calculate(i) the probability of any
When throwing a die once, find the probability that the outcome is(i) an even integer;(ii) at least 5;(iii) at most 3;(iv) divisible by 3.
For the experiment of tossing two coins, find the probability that at least one head appears using the sample space Ω = {0, 1, 2}, suggested by D’Alembert (see Example 2.4).(Hint: Keep in mind
29. In a study of platelet production, 16 rats were put at an altitude of 15,000 feet, while another 16 were kept at sea level (Rand, K., Anderson, T., Lukis, G., and Creger, W., “Effect of hypoxia
28. An experiment has been devised to test the hypothesis that an elderly person’s memory retention can be improved by a set of “oxygen treatments.” A group of scientists administered these
27. Suppose, in Problem 23, that there has been some controversy about the assumption of no interaction between gasoline and additive used. To allow for the possibility of an interaction effect
26. A study was made as to how the concentration of a certain drug in the blood, 24 hours after being injected, is influenced by age and gender. An analysis of the blood samples of 40 people given
25. A researcher is interested in comparing the breaking strength of different laminated beams made from 3 different types of glue and 3 varieties of wood. To make the comparison, 5 beams of each of
24. Suppose in Problem 6 that the 10 people placed on each diet consisted of 5 men and 5 women, with the following data.(a) Test the hypothesis that there is no interaction between gender and
23. An experiment was devised to test the effects of running 3 different types of gasoline with 3 possible types of additive. The experiment called for 9 identical motors to be run with 5 gallons for
22. Three different washing machines were employed to test four different detergents.The following data give a coded score of the effectiveness of each washing.(a) Estimate the improvement in mean
20. The following data refer to the number of deaths per 10,000 adults in a large Eastern city in the different seasons for the years 1982 to 1986.(a) Assuming a two-factor model, estimate the
19. Astudy has been made on pyrethrum flowers to determine the content of pyrethrin, a chemical used in insecticides. Four methods of extracting the chemical are used and samples are obtained from
18. If xij = ai + bj , show that mn m =a+m; i=1 j=1 i=1 j=1
17. If xij = i + j2, determine j=1 (a) (6)
16. For data xij , i = 1, . . . ,m, j = 1, . . . , n, show that m =xilm = xjln x = i=1 j=1
15. Test the hypothesis that the following three independent samples all come from the same normal probability distribution. Sample 1 Sample 2 Sample 3 35 29 44 37 38 52 29 34 56 27 30 30 32
14. A nutritionist randomly divided 15 bicyclists into 3 groups of 5 each. The first group was given a vitamin supplement to take with each of their meals during the next 3 weeks. The second group
13. Five servings each of three different brands of processed meat were tested for fat content. The following data (in fat percentage per gram) resulted.(a) Does the fat content differ depending on
12. An emergency room physician wanted to know whether there were any differences in the amount of time it takes for three different inhaled steroids to clear a mild asthmatic attack. Over a period
11. A study of the trunk flexor muscle strength of 75 girls aged 3 to 7 was reported by Baldauf, K., Swenson, D., Medeiros, J., and Radtka, S., “Clinical assessment of trunk flexor muscle strength
10. Plasma bradykininogen levels are related to the body’s ability to resist inflammation.In a 1968 study (Eilam, N., Johnson, P. K., Johnson, N. L., and Creger, W.,“Bradykininogen levels in
9. The following data relate to the ages at death of a certain species of rats that were fed 1 of 3 types of diet. Thirty rats of a type having a short life span were randomly divided into 3 groups
8. In the one-factor analysis of variance model with n observations per sample, let S2 i , i = 1, . . . ,m denote the sample variances for the m samples. Show that m SSw = (n - 1)S i=1
7. In a test of the ability of a certain polymer to remove toxic wastes from water, experiments were conducted at three different temperatures. The data below give the percentages of the impurities

Showing 1300 - 1400 of 7137

First
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
Last

Services

Sitemap
Fun
Definitions
Become Tutor
Study Help Categories
Recent Questions
Expert Questions

Company Info

Security
Copyrights
Privacy Policy
Terms & Conditions
SolutionInn Fee
Scholarship

Get In Touch

About Us
Contact Us
Career
Jobs
FAQ
Campus Ambassador

Secure Payment

Download Our App

© 2024 SolutionInn. All Rights Reserved