. The probability that two specified letters will be placed in the correct envelopes is 1[n(n – 1)]. Theprobability that none of the othern – 2 letters will then be placed in the correct envelopes is qn-2.Therefore, the probability that only the two specified letters, and no other letters, will be placedin the correct envelopes is1-¶n-2. It follows that the probability that exactly two of the nп(п — 1)letters will be placed in the correct envelopes, without specifying which pair will be correctly placed,

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Answered: . The probability that two specified…

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

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...

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The resumes of two male applicants for a college teaching position in Engineering Data Analysis are placed in the same file as the resumes of two female applicants. Two positions become available, and the first, at the rank of assistant professor, is filled by selecting one of the foUr applicants at random. The second position, at the rank of instructor, is then filled by selecting at random one of the remaining three applicants. Using the notation M,F1, for example, to denote the simple event that the first position is filled by the second male applicant and the second position is then filled by the first female applicant. Find the probability that the position of assistant professor is filled by a male applicant. O 1/3 O 1/6 O 2/3 O 1/2
In the spreading of a rumour, each person who is told the rumour has probability p of believing it and only those who believe the rumour will pass it on. Assume that all individuals behave independently of each other and the rumour is never told to someone who has already heard it. Assume also that each person who is told the rumour and believes it on the n-th day will pass the rumour on to two other people on the (n + 1)-st day. Suppose that the rumour originates on day 1 from a single individual telling the rumour to two other people. Let Zo = 1 and for n ≥ 1 denote by Z₁, the number of people who are told the rumour on day n and believe it. (i) Explain why (Zn)nzo is a branching process. (ii) Find the probability mass function and the probability generating function of Z₁. (iii) What is the probability that the spread of the rumour ultimately ceases? (iv) For what values of p is the spread of the rumour certain to cease eventually? (v) For n ≥ 1 show that P(Zn = 2") = p2¹ +22+ +2″
there are no pairs of consecutive integers in the winning combination (i1,..., ix) if and only if (j1, ..., jk) has no repeats. The total number of winning combinations is In part k (c), we computed the number of winning combinations with no repeats among (j1,.. ,jk) to be n – k+1 So, the probability of no consecutive integers is k
of Suppose a child is born to two parents who are cystic fibrosis carriers. Construct a chart of gene combinations to find the probability that the child will neither have nor carry the disease. P(neither have nor carry the disease) = (Type an integer or a simplified fraction.) of of y of of C Time Remaining: 01-0637
. If A is an arbitrary event, then show that P(A) = 1- P(A).
There are two coins, one with probability p1 of getting Heads and the other with probabilityp2 of getting heads. One of the coins is randomly chosen (with equal probabilities for thetwo coins). It is then flipped n ≥ 2 times. Let X be the number of times it lands Heads.(a) Find the PMF of X. Your answer should be a formula involving n, k, p1,and p2. (Hint: Law of Total Probability.)(b) What is the distribution of X if p1 = p2?
We are at a city named Freezo, close to the North Pole, where the days are either "warm" or "cold" (with temperature either above of below freezing). Assume that if we have a cold day today, then tomorrow will be also be cold with probability p (and warm with probability 1 – p), and if today is warm then tomorrow will be warm with probability q (and cold with probability 1 – q). Here p and q are some numbers in the interval (0, 1). Consider a matrix p 1- A = (8a) Prove that 1 = 1 is an eigenvalue of A (for any choice of p and q). (8b)| ] Set p = 3/4 and q = 1/2. Find the eigenvector of A corresponding to 1 = 1, such that x + y = 1. (8c) e It turns out that the numbers x and y that you found in (8b) represent the long-term probability that any given day will be cold and warm, respectively. We are planning a 30-day event "Snowtopia" in 2031 that requires at least 18 cold days during the event. Is Freezo a suitable place to host this event? Explain why.
Q6 At all times, an urn contains N balls- some white balls and some black balls. At each stage, a coin having probability p, 0 < p < 1, of landing heads, is flipped. If heads appear, then a ball is chosen at random from the urn and is replaced by a white ball; if tails appear, then a ball is chosen from the urn and is replaced by a black ball. Let Xn denote the number of white balls in the urn after the n stages. Q6(i.) Is {Xn, n 2 0} a Markov Chain? If so, explain why. Q6(ii.) Compute the transition probabilities Pij: (Hint: At some stage, suppose there are i white balls (and N – i black balls). Think how you can get i or i – 1 or i +1 white balls in the next stage.) Q6(iii.) Let N 2. Then the states are 0, 1, 2 and the transition probability matrix is given by 를 + 올 끌 Find the proportion of the time in each state.

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