A First Course in Probability (10th Edition)
10th Edition
ISBN: 9780134753119
Author: Sheldon Ross
Publisher: PEARSON
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Chapter 1, 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 equal. How many have exactly 2 equal digits.
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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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- Total marks 14 4. Let X and Y be random variables on a probability space (N, F, P) that take values in [0, ∞). Assume that the joint density function of X and Y on [0, ∞) × [0, ∞) is given by f(x, y) = 2e-2x-y Find the probability P(0 ≤ X ≤ 1,0 ≤ y ≤ 2). (ii) spectively. [6 Marks] Find the the probability density function of X and Y, re- [5 Marks] 111) Are the X and Y independent? Justify your answer! [3 Marks]arrow_forwardTotal marks 17 4. Let (,,P) be a probability space and let X : → R be a ran- dom variable that has Gamma(2, 1) distribution, i.e., the distribution of the random variable X is the probability measure on ((0, ∞), B((0, ∞))) given by (i) dPx(x) = xex dx. Find the characteristic function of the random variable X. [8 Marks] (ii) Using the result of (i), calculate the first three moments of the random variable X, i.e., E(X") for n = 1, 2, 3. Using Markov's inequality involving E(X³), (iii) probability P(X > 10). [6 Marks] estimate the [3 Marks]arrow_forward1. There are 8 balls in an urn, of which 6 balls are red, 1 ball is blue and 1 ball is white. You draw a ball from the urn at random, note its colour, do not return the ball to the urn, and then draw a second ball, note its colour, do not return the ball to the urn, and finally draw a third ball, note its colour. (i) (Q, F, P). Describe the corresponding discrete probability space [7 Marks] (ii) Consider the following event, A: At least one of the first two balls is red.arrow_forward
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