5. Suppose you have a fair die. (Namely, when the die is tossed, it is equally likely that any of the six numbers-1, 2, . . . , 6–will appear.) You are playing a game where, if a 1 or 2 appears, you win $2. If a 3 or 4 appears, you win $1. If a 5 appears, you do not win or lose any money. Finally, if a 6 appears, you lose $6. (a) Find the expectation of this game. (b) Find the variance of this game. (c) If the game is played thirty times what is the probability of winning moer than $10? Use the central limit theorem to approximate this probability. Use the table of values for the standard normal distribution on the last page of this exam. Try not to round anything until you get to the table.

5. Suppose you have a fair die. (Namely, when the die is tossed, it is equally likely that any of the six numbers-1, 2, . . . , 6–will appear.) You are playing a game where, if a 1 or 2 appears, you win $2. If a 3 or 4 appears, you win $1. If a 5 appears, you do not win or lose any money. Finally, if a 6 appears, you lose $6. (a) Find the expectation of this game. (b) Find the variance of this game. (c) If the game is played thirty times what is the probability of winning moer than $10? Use the central limit theorem to approximate this probability. Use the table of values for the standard normal distribution on the last page of this exam. Try not to round anything until you get to the table.

Name: 5. Suppose you have a fair die. (Namely, when the die is tossed, it i
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Description: 5. Suppose you have a fair die. (Namely, when the die is tossed, it is equally likely that any of the sixnumbers-1, 2, . .6–will appear.) You are playing a game where, if a 1 or 2 appears, you win $2. If a3 or 4 appears, you win $1. If a 5 appears,

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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A casino is offering a gamble to customers. A single, fair six-sided die is rolled. If a player rolls 1,2, or 3, the payoff is the number rolled in dollars. If the player rolls 4,5, or 6, they get to draw from a standard 52-card deck of cards and gets paid the number on the card in dollars (kings, queens, jacks, and aces are each worth $10). What price to play makes this a fair gamble?
Consider a new card game between 2 players: Jim (player 1) and Angella (player 2) Jim is dealt two cards : ♡8 and ♠️9 . Angella is also dealt two cards: ♢6 and ♣️6 . Now, each of the players will play 1 card both at the same time. The payoff of Jim is 4 points if he plays a card of opposite color (red/black) than Angella, and otherwise his payoff is 10 points. The payoff of Angella is 3 points if the difference of the already played card numbers is greater than 4, otherwise her payoff is 2 points. 1. Find the Best Responses for Angella.2. Find all the Nash Equilibriums of the game (if any).
10. There are two players, Coin and Die. Coin has a coin, and Die has a single six-sided die. The rules are: When it is Coin's turn, he or she flips the coin. If the coin turns up heads, Coin wins the game. If the coin turns up tails, it is Die's turn. When it is Die's turn, he or she rolls the die. If the die turns up 1, Die wins. If the die turns up 6, it is Coin's turn. Otherwise, Die rolls again. When it is Coin's turn, the probability is 1/2 that Coin will win and 1/2 that it will become Die's turn. When it is Die's turn, the probabilities are 1/6 that Die will roll a 1 and win 1/6 that Die will roll a 6 and it will become Coin's turn, and 2/3 that Die will roll a 2, 3, 4, or 5 and have another turn. To describe this process as a Markov chain, we define four states of the process: State 0 : Coin has won the game. State 1: It is Coin's turn. State 2: Die has won the game. State 3: It is Die's turn. (a) Give the transition matrix P for this Markov chain. (Hint: I have done the first…
Suppose you play a game with a biased coin. You play each game by tossing the coin once. and . If you toss a head, you pay $10. If you toss a tail, you win $11. if you play this game many times, will you come out ahead? (i.e. will you gain extra money from playing this game?) (Be sure to use fractions in the table.)
In a gambling game, I roll two dice and win $1 if the sum of my two rolls is largerthan 7. I lose $1 if the sum of my two rolls is smaller than 7. In any other cases, Idon’t lose or win anything. I am going to repeat this game 72 times. c) Now make a box model for the number of times I win, including the values onthe tickets, the number of repeats of each ticket, and how many draws we willmake with replacement from the box. The average of the tickets in the box from (c) is ___________ and the SD is___________.
5. Divide and conquer. You're again facing your friend, Brenda, in a "candy-off” game involving a pile of 100 caramels and the winner's prize: one peppermint patty. (Like before, you still LOVE peppermint patties!). On their turn, each player can remove a
Consider the game of Let’s Make a Deal in which there are three doors (numbered 1, 2, 3), one of which has a car behind it and two of which are empty (have “prizes”). You initially select Door 1, then, before it is opened, Monty Hall tells you that Door 3 is empty (has a prize). You are then given the option to switch your selection from Door 1 to the unopened Door 2.
Suppose that you are offered the following "deal." You roll a six sided die. If you roll a 6, you win $15. If you roll a 4 or 5, you win $5. Otherwise, you pay $9.
One thousand raffle tickets are sold at $1 each. Three tickets will be drawn at random (without replacement), and each will pay $201. Suppose you buy 5 tickets. (A) Create a payoff table for 0, 1, 2, and 3 winning tickets among the 5 tickets you purchased. (If you do not have any winning tickets, you lose $5, if you have 1 winning ticket, you net $196 since your initial $5 will not be returned to you, and so on.) (B) What is the expected value of the raffle to you? (A) Let X be the random variable for the amount won on a single raffle ticket. Create a payoff table for 0, 1, 2, and 3 winning tickets among the 5 tickets you purchased. Xi-$5 $196 $397 $598
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