Consider the following parlor game between two players. It begins when a refereeflips a coin, notes whether it comes up heads or tails, and then shows this result toplayer 1 only. Player 1 may then (i) pass and thereby pay $15 to player 2 or (ii) bet. Ifplayer 1 passes, the game is terminated. However, if he bets, the game continues, inwhich case player 2 may then either (i) pass and thereby pay $15 to player 1 or (ii)call. If player 2 calls, the referee then shows him the coin; if it came up heads, player2 pays $30 to player 1; if it came up tails, player 2 receives $30 from player 1.(1) Is there a saddle point in this game? Type "Yes" or "No".A(2) What is the value of the game?A/

Answered: Consider the following parlor game…

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 casino game of craps is played as follows: Step 1: The player rolls a pair of standard 6-sided dice and takes their sum a. If the sum is 7 or 11, then the player wins and the game is over b. If the sum is 2, 3, or 12, then the player loses (this is called “crapping out”) andthe game is over c. If the sum is anything else, then we record the sum (lets call it “X”) and continueto the next step Step 2: The player then rerolls the dice and takes their sum a. If the sum is X, the player wins and the game is over b. If the sum is 7, the player loses and the game is over c. If the sum is anything else, repeat step 2. Now suppose that you notice something odd - one of the two dice isn’t balanced that well, and always comes up in the range 2-5 (with equal probability) but never 1 or 6. 1. For each number between 2 and 12, what is the probability of rolling the dice so that they sum to that number?
2. Janelle plays a game in which she chooses a card at random from a standard deck of 52 cards. If the card selected is a face card, she wins $2; if it is an ace, she wins $5; if it is an even numbered card, she wins $1; but if it is an odd numbered card, she loses $3. Find her expected gain for this game.
The Dice game of “Pig” can be played with the following rules. Roll two six-sided dice. Add the face values together. Choose whether to roll the dice again or pass the dice to your opponent. If you pass, then you get to bank any points earned on your turn. Those points become permanent. If you roll again, then add your result to your previous score, but you run the risk of losing all points earned since your opponent had rolled. Continue to roll as much as you want. However, once a “1” comes up on either die, your score is reduced to 0, leaving you only with points that you have previously "banked." Furthermore, you must pass the dice to your opponent. The first person to 100 points is the winner. When a player rolls two dice, the possible outcomes are as follows: Die #2 Roll Roll a 1 Roll a 2 Roll a 3 Roll a 4 Roll a 5 Roll a 6 Die #1 Roll Roll a 1 0 0 0 0 0 0 Roll a 2 0 4 5 6 7 8 Roll a 3 0 5 6 7 8 9 Roll a 4 0 6 7 8 9 10 Roll a 5 0 7 8 9 10 11 Roll a 6 0 8 9 10…
1
Consider the following game. You are to toss three fair coins. If three heads turn up, your friend pays you $40. Otherwise, you must pay him $5. What is your expected gain or loss? 0.525 a. 0.625 Ob. 0.875 C. Od. 1235 0.125 Oe.
In a game of chance you will be given $6.00 every time you roll a 2 with a balanced die.a) What is your expected payoff?b) How much should you pay when you roll a 1,3,4,5,or 6 to make the gameequitable? A game is equitable if the mathematical expectation of a player is 0.
Charlie buys a one-year term home insurance policy for $p, that will pay $200,000 in the event of a major catastrophe and $35,000 in the event of a minor catastrophe. Charlie's home has a 0.2% chance of a major catastrophe and a 0.7% of a minor catastrophe during the one year term. What did the insurance company charge Charlie for this policy if the company expected to break even?
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.)
In craps which of the following scenarios is least likely? (Remember, for each roll you're rolling two dice, so rolling a 7" means any combination of dice that adds to 7.) A. Roll an 8 on the first roll, and an 8 on the second roll to win the game. B. Roll a 4 on the first roll, and a 7 on the second roll to lose the game. C. Roll a 7 on the first roll to win the game. D. Roll a 6 on the first roll, and a 6 on the second roll to win the game. O E. Roll a 2, 3, or 12 on the first roll to lose the game.
1. The casino game of chuck-a-luck seems quite simple and that it would be a pretty good bet. In this game, three dice are rolled (usually inside a wire cage). You can bet a set amount, let's say $5, on a specific number 1-6. You win the amount of your bet for each appearance of your number on the dice. If the number does not appear, you lose your bet. For example, let's say you bet $5 on the number 3. If it appears on exactly one die, you win $5 (and keep your bet) with probability 75/216. If it appears on exactly two dice, you win $10 (and keep your bet) with probability 15/216. If it appears on all three dice, you win $15 (and keep your bet) with probability 1/216. If it appears on none of the dice, you lose your original $5 bet. Construct the probability distribution for chuck-a-luck and determine your expected winnings on a $5 bet.
1. Consider a game played by rolling a set of three 3-sided dice one time and adding the values from the dice together to form a sum. The game costs $7 to play once. You don't get any prize for rolling a sum of 5 or 7, you get $12 for a sum of 4 or 8, and you get $37 for a sum of 3 or 9. If the game is "fair," what is the prize for rolling a 6? (Hint: form a probability distribution to help you find the missing value and recall what "expected value" means in the context of a probability distribution.)
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?

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