(2) Consider the following game of chance. You pay $30 to enter the game. First, you...
Consider a pay-to-play game which involves flipping a coin three (3) times. The payout for the game depends on the number of heads obtained in the three coin flips. Let the discrete random variable X represent the number of heads. a) What is the probability P{X = k} associated with each value k of the random variable? b) Suppose that the game has a payout of X^2 dollars. What is the minimum amount that should be charged for admittance (player...
For this question, you will flip fair coin to take some samples and analyze them. First, take any fair coin and flip it 12 times. Count the number of heads out of the 12 flips. This is your first sample. Do this 4 more times and count the number of heads out of the 12 flips in each sample. Thus, you should have 5 samples of 12 flips each. The important number is the number of heads in each sample...
Suppose that you believe there is a 30% chance that the coin in your hand is biased 80% in favor of heads, and a 70% chance that it’s fair. You flip it twice and get heads both times. What should you now believe is the probability that the coin is fair?
2. Mysterioso the Magician is walking down the street with a box containing 25 identical looking coins: 24 are fair coins (which flip heads with probabilty 0.5 and tails with probability 0.5) and one is a trick coin which always flips heads. Renata the Fox skillfully robs Mysterioso of one of the coins in his box (chosen uniformly at random). She decides she will flip the coin k times to test if it is the trick coin (a) What is...
Suppose you can place a bet in the following game. You flip a fair coin (50-50 chance it lands heads). If it lands heads, you get 4 dollars, if it lands tails, you pay 1 dollar. This is the only bet you can make. If you don't make the bet you will neither gain nor lose money. What is the utility for you of the coin landing tails if you make the bet (assume utility is dollars)?
Suppose you can place a bet in the following game. You flip a fair coin (50-50 chance it lands heads). If it lands heads, you get 4 dollars, if it lands tails, you pay 1 dollar. This is the only bet you can make. If you don't make the bet you will neither gain nor lose money. Should you place the bet?
Boris and Natasha agree to play the following game. They will flip a (fair) coin 5 times in a row. They will compute S = (number of heads H – number of tails T). a) Boris will pay Natasha S. Graph Natasha’s payoff as a function of S. What is the expected value of S? b) How much should Natasha be willing to pay Boris to play this game? After paying this amount, what is her best case and worst...
Consider the setting where you first roll a fair 6-sided die, and then you flip a fair coin the number of times shown by the die. Let D refer to the outcome of the die roll (i.e., number of coin flips) and let H refer to the number of heads observed after D coin flips. (a) Suppose the outcome of rolling the fair 6-sided die is d. Determine E[H|d] and Var(H|d). (b) Determine E[H] and Var(H).
Problem 3. You play a game where you first choose a positive integer flip a fair coinn times. You win a prize if you get exactly 2 heads. How should you choose n to maximize your chance of winning? What is the chance if winning with optimal choice of n? There are two equally good choices for the best n. Find both n and then an
Suppose you make a dollar bet on a game in which there is a 1 in 5 chance to win. If you win, you win two dollars. On average, you will lose playing this game and each play costs you _______ cents. If you play 200 times, you can expect to lose around _______ dollar .You play roulette betting one dollar on the number 5 each time. The bet pays 35 to 1. You have a 1 in 38 chance to...