Q3)Consider an infinite-length path that consists of discrete segments. There is a person that ismoving on this path according to a simple rule: At each turn, the person throws a (possibly unfair)dice. Suppose the output of the dice is k, then the person jumps k segments ahead and proceedswith throwing the dice again. For instance, if the person is at the n-th segment and the outputof the dice is 3, the person jumps 3 segments and steps on the (n+3)-th segment. We do nothave any idea about the starting point of the person; assume that the person is coming from minusinfinity and going to plus infinitya) Suppose the dice is unfair, and the person always throws the same fixed k, k € {1, 2, 3, 4, 5, 6},with probability 1. You mark a distinct segment on the path (for example segment n, for somefixed constant n). What is the probability that the person steps on this segment? (Hint: yourresult must be in terms of k)b) Consider the setup in part (a) with a fair dice i.e. all outcomes are equally likely. What isthe probability that the person steps on this segment? (Hint: your result must not be in terms ofk. Think about the average speed of this person)

a) Suppose the person always throws the same fixed number k. The probability of landing on the…

Answered: Consider an infinite-length path that…

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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1. Consider the gambler's ruin problem. Alice starts with £a; her opponent Bob starts with £(ma). In each round, Alice wins £1 from Bob with probability p, or loses £1 to Bob with probability q = 1 - p. Alice ruins if she is reduced to £0; Bob ruins if he is reduced to £0, which means Alice would have £m. Let X denote the amount of money in pounds that Alice has after n rounds. Consider the stochastic process (✗n). pri+1ri+gri-1 = 0 for i = 1, 2,..., m — 1, Assuming p, solve the linear difference equation in part (c), to show that the ruin probability is pm - pa ra pm 1 ' where you should identify p in terms of p and q.
4. A college wants to investigate whether students perform better on mornings that they have hada nutritious breakfast compared to mornings that they have not.You randomly recruit 100 students and tell each of them to come to the lab in two different mornings:each morning they will be offered some breakfast and will take a test. Each student will receivethe light breakfast one time and the nutritious breakfast the other time (some will receive the lightbreakfast first, the others will receive the nutritious breakfast first, and the second time they switch). The average difference in a student’s score after eating the nutritious breakfast and their score aftereating a light breakfast is 4.5, and the standard deviation of this difference is 3.6. Thus we aredoing students nutritious breakfast scores minus the light breakfast scores. a. Construct a 95% confidence interval for the difference in the average score of students afterhaving a nutritious breakfast and the average score of…
This student never eats the same kind of food for 2 consecutive weeks. If she eats in a Korean restaurant one week, then she is four times as likely to have Turkish as French food the next week. If she eats in a Turkish restaurant one week, then she is equally likely to have Korean as French food the next week. If she eats in a French restaurant one week, then she is five times as likely to have Korean as Turkish food the next week. Assume that state 1 is Korean and that state 2 is Turkish, and state 3 is French. Find the transition matrix for this Markov process. ... ... P = ... ...
Ten times a coin is flipped, and each time it's either heads or tails. Is there a limit to the number of outcomes that have less than four tails?
5. Given 20 different types of inputs to a program, in how many ways can 8 of them be selected if (a) order does not matter. (b) order matters.
“ You are dealt five cards from a standard deck of 52 cards. How many of these handshave at least one queenandat least one spade. ”He breaks this into two cases. In the first case, he looks at hands which contain the queen of spadesQ♠. There is 1 way of choosing the Queen of Spades, and51C4ways of choosing the other fourcards. In the second case, he will start by choosing a non-spade queen (3 ways) and a non-queenspade (12 ways). There are 50 cards remaining, so he also throws away the Queen of Spades sothat he doesn’t overlap case 1. This gives49C3ways of choosing the last three cards. He concludesthat the answer is51C4+ (3·12·49C3). His answer is incorrect. Explain why, and give the correctanswer.
Social Night Problem 2: In the above diagram, on the left, we have some girls and on the right, we have some guys. If there is an edge between a girl and a guy, it means both of them are attracted to each other. Yes, a person can have multiple crushes. In this scenario, what is the maximum number of couples that can result from the above diagram given that nobody wants to end up with someone they don't like? A person would rather stay single than end up with someone they don't like. What matching algorithm are you using for this problem? Show each step of your algorithm precisely.
Use generating functions to solve the following problem: How many ways can n distinct 6-sided dice show a sum of 24? Your solution should include constraints on n (i.e. if n= 100, the answer is 0 since the minimum possible sum is 100, and if n= 1 the answer is also zero).
Consider a game with n ∈ N participants which are somehow ordered (P1, P2, ..., Pn). The game starts with the first numbered player (P1) tosses a fair coin until the first “Tail” appears. By x1 we denote the number of flips made by P1. Player P1 is eliminated from the game if x1 < y. In this case, the second player tosses a fair coin until the first “Tail” appears. Similarly, by x2 we denote the number of flips made by P2. Player P2 is eliminated from the game if x1 + x2 < y. The game proceeds this way until either the total number of coin flips attains y, or all players are eliminated. All those players who are not eliminated at the end of the game are winners. Find the expected number of tosses for a player i (E(xi)) assuming there is no limit on the total number of flips (y).
I need help on both questions. Thank you in advance!
"Question 3 (MaC)". (i) John has 5 toy cars and 4 toy buses that are all different from each other. John arranges these 9 toys in a line. Find the number of possible arrangements if: (a) (a) the cars and buses are arranged alternately. (B) the buses are all next to each other. (b) John randomly chooses 4 toys to take on holiday with him. Find (a) in how many ways John can choose the 4 toys. (B) the probability that his choices will include both his favourite car and his favourite bus. | (ii) Consider the function f(x) = x" +1 (a) Evaluate lim x→® x* +1 (b) Find the stationary point of the curve y = f(x) and determine its nature. (c) Find the point(s) of inflection of the curve y = f(x). (d) Sketch the curve y = f(x) showing all the features found above. Using your graph, find all values of k for which the equation kx² = 1-k has exactly one solution. (e)
I conduct an online study to determine whether rewards reduce procrastination. All participants are asked to complete a task sometime before a deadline (10 days from now). The computer randomly assigns participants to one of three groups, and emails them their instructions; one group receives a reward as soon as they do the task (contingent reward); a second group receives rewards as soon as they finish the task as long as it is before the deadline (non-contingent reward) and the third group does not receive rewards at all (no reward). Three participants in the no reward condition do not complete the task, and so are dropped from analyses. The dependent variable is the amount of time (in days) it takes participants to complete the task. The JAMOVI output is attached. Post-hoc Tukey tests show that... (select all that apply) ANOVA ANOVA - Time Reward Residuals Assumption Checks Homogeneity of Variances Test (Levene's) F df1 df2 0.056 Statistic 0.964 Sum of Squares df Mean Square F…

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