At time 0, John has $2. At times 1, 2,..., he independently plays a game in whichhe bets $1. With probability p= 0.45, he wins the game and with probability 1game. His goal is to increase his capital to $3, and as soon as he does, the game is over. The game is alsop = 0.55, he loses theover if his capital is reduced to zero. Construct an absorbing Markov chain and answer the followingquestions.• What is the expected duration of the game?• What is the probability that he goes broke?

State 0 is John's capital is $0 (Broke).State 1 is John's capital is $1.State 2 is John's capital is…

Answered: At time 0, John has $2. At times 1, 2,…

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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One option in a roulette game is to bet $20 on red. (There are 18 red compartments, 18 black compartments, and two compartments that are neither red nor black.) If the ball lands on red, you get to keep the $20 you paid to play the game and you are awarded $20. If the ball lands elsewhere, you are awarded nothing and the $20 that you bet is collected. Find the expected payback for this roulette game if you bet $20 on red. The expected payback is $ (Round to the nearest cent as needed.)
Roulette is a casino game that involves players betting on where a ball will land on a spinning wheel. An American roulette wheel has 38 numbered slots — half of the slots from 1 to 36 are red and the other half are black. Slots 00 and 00 are both green. Suppose that a player bets $1 on a single slot. If the ball lands in their slot, the player gets their initial $1 back plus a payout of $35. If the ball doesn't land in their slot, they lose their $1 bet. Let X= the player's net gain from a $1 bet on a single slot. Here is the probability distribution of X: Find the expected value of a player's net gain on a $1 bet on a single slot.You may round your answer to the nearest thousandth.
The St. Petersburg paradox is a coin flip game, where a fair coin is tossed successively, until heads occurs. If the first toss is heads, you win $1. If the first one is tails and the second one is heads, you win $2. If the first heads occurs in the third toss, you win $4 and so forth. The amount of money you win is doubled with each coin flip you survive. In the eighteenth century it was believed that the fair price for participating in such a game would be the expected wealth you gain. Show that the expectation value of the St. Petersburg game is infinite.
In an experimental study, researchers had each of their participants bet on each game of a professional football season. In the contingency table below is some information from a random sample of 100 bets from this study placed on the Columbus Crush (picking them to win) during the last 14 games of the season (the Crush had 7 wins and 7 losses over that period). The table indicates, for each bet placed on the Crush, whether or not the team won and how the participant who placed the bet wagered the following week. Each bet is classified according to two variables: result of picking the Crush ("Crush won" or "Crush lost") and bet placed the following week ("Picked Crush to win" or "Picked Crush to lose"). In the cells of the table are the respective observed frequencies, and three of the cells also have blanks. Fill in these blanks with the frequencies expected if the two variables, result of picking the Crush and bet placed the following week, are independent. Round your answers to two…
Leah is flying from Moncton to Vancouver with connection in Montreal. The proba- bility that her first flight leaves on time is 0.15. If the flight is on time, the probability that her luggage will make the connecting flight in Montreal is 0.95, but if the first is Practice q # 25 p.197 delayed, the probability that the luggage will make it is only 0.65. chap 7 Are the first flight leaving on time and the luggage making the connection independent events? Explain. What is the probability that Leah's luggage arrives in Vancouver with a. b. her?
A trader invented a strategy where he can either win or lose a certain amount of money every day (say, k dollars) with equal probability, independently of all other days. (In fact, such a strategy is not very difficult to implement - buy a very liquid asset whose price moves without major jumps and liquidate the position as soon as you are making or losing k dollars). He then made the following statement: “If I use my strategy every day for the next 400 days, my chances of staying within 10,000 dollars of the original account balance are equal to 1/3.(a) What is k (approximately)?(b) What are his chances (approximately) of making over 20,000 with this strategy in 400 days?
Throughout a month (30 days), everyday you play a certain game exactly once.On the first day you deposit 1000 coins and everyday you bet any positive amount (not exceeding the deposit) and at the end of the game,your bet is tripled (with probability 4/9) and added to your deposit or lost (with probability 5/9), independently each day.If on some day, you bet all your deposit and lose the game, then you are bankrupt and game ends.How much should you bet on each day?
In an experimental study, researchers had each of their participants bet on each game of a professional football season. In the contingency table below is some information from a random sample of 100 bets from this study placed on the Columbus Crush (picking them to win) during the last 14 games of the season (the Crush had 7 wins and 7 losses over that period). The table indicates, for each bet placed on the Crush, whether or not the team won and how the participant who placed the bet wagered the following week. Each bet is classified according to two variables: result of picking the Crush ("Crush won" or "Crush lost") and bet placed the following week ("Picked Crush to win" or "Picked Crush to lose"). In the cells of the table are the respective observed frequencies, and three of the cells also have blanks. Fill in these blanks with the frequencies expected if the two variables, result of picking the Crush and bet placed the following week, are independent. Round your answers to two…
3) At time 0, Elon Musk (the cool guy these days) has $2. At times 1, 2,..., he independently plays a game in which he bets $1. With probability p = 0.49, he wins the game and with probability 1 - p = 0.51, he loses the game. His goal is to increase his capital to $3, and as soon as he does, the game is over. The game is also over if his capital is reduced to zero. Construct an absorbing Markov chain and answer the following questions. • What is the expected duration of the game? • What is the probability that he goes broke?
Diane has decided to play the following game of chance. She places a $1 bet on each repeated play of the game in which the probability of her winning $1 is .4. She has further decided to continue playing the game until she has either accumulated a total of $3 or has lost all her money. What is the probability that Diane will eventually leave the game a winner if she started with a capital of $1? Of $2? capital of $1 capital of $2
Alyssa and Gabriel play a game. Alyssa rolls a die. If she rolls a 1, 2, or 3, Gabriel gives away $3. If Alyssa rolls a 4 or 5, he gives her $2. However if she rolls a 6, she gives him $22. Determine Alyssa's expectation and determine Gabriel's expectation

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