Suppose there are k students in a class, where2 ≤ k ≤ 365. What is the probability that at least two students have the same birthday? Here,by the same birthday, we mean the same day of the same month, but not necessarily of the sameyear. Moreover, we make the following assumptions:(1) No twins in the class;(2) Each of the 365 days is equally likely to be the birthday of anyone in the class;(3) Anyone born on Feb. 29 will be considered as on March 1.

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Answered: Suppose there are k students in a…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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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 1- A = (8a) Prove that A = 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 v = of A corresponding to A= 1, such that x + y = 1. (8c) 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.
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″
Three distinct numbers between 10 and 25 (inclusive) are chosen at random. What is the probability that the numbers are all composite numbers?
If, over a certain period of time, rain falls at random and all things equal, on about 11 days in every 30, find the probability that the first 4 days of a given week will be fine and the remainder wet.
In a certain problem if A tells the truth 4 times out of 5 times and B tells the truth 3 times out of 4 times. The probability that both expressing the same fact contradict to each other is?.
(1) On their first date, Kelly asks Mike to guess the date of her birth, not including the year. d) If Kelly asks Mike to guess her age, and Mike's guess is too high by 15 years, what is the probability that Mike and Kelly will have a second date?
each of the natural numbers from 1 to 15 inclusive is written on one card, and different numbers are written on the different cards. We randomly select four cards. What is the probability that the smallest number of selected cards is 8 or the largest of them is 12?
A surprising result that appears in many Elementary discussions on probability is called the birthday problem. The question simply stated is that if we poll a certain number of people, what is the probability that at least two of those people were born on the same day of the year? To solve this problem, use the formula for computing the probability of a complement of a event. It is clear that P(duplication of some birthdays)=1-P(n I duplications). Assume that there are 4 people. Find the probability that at least 2 of the 4 people were born on the same day of the year. Do not consider February 29th in leap year when finding the probability.
If the student attends class on a certain Friday, then he is five times as likely to be absent the next Friday as to attend. If the student is absent on a certain Friday, then he is three times as likely to attend class the next Friday as to be absent again.
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.

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