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- QUESTION 24 A deck of 52 playing cards consists of 4 suites: clubs, spades, hearts, and diamonds. Each suite consists of 13 cards. Draw one from a deck of cards. Let A=a card "2" is drawn, B=a "diamond" card is drawn. Which is the value of P(A|B), i.e., {probability that the card drawn is "2" given that we have known it is "diamond"} ? 1/52 1/4 1/13 16/52QUESTION 26 A deck of 52 playing cards consists of 4 suites: clubs, spades, hearts, and diamonds. Each suite consists of 13 cards. Draw one from a deck of cards. Let A=a card "2" is drawn, B=a "diamond" card is drawn. Which is the value of P(A and B), i.e., {probability that the card drawn is "diamond" and "2"} ? 1/52 4/52 13/52 16/52The injured football player Bad news everyone! There is 1 second left in the game, and Tom Brady has injured himself. The matrices below depict the relative probabilities of winning givenan offensive and a defensive play call. (The row player is the New England Patriots and the column player is the opponent.) How much has the all star's home team probability of winning decreased due to the injury? Pass uny Patriots D Pass .4, .6 D Run .9,.1 .8,.2 .5,.5 Pass Run Opponent D Pass D Run .06, .94 .32, .68 .8,.2 .5,.5
- Find the perfect Bayesian equilibria in the following two-person game: B. Chance 1/4 U 3/4 2 L \R (:) (?) (3) (6) () (:) (:) (*) 8n people guess an integer between 1 and 100, and the winner is the player whose guess is closest to the mean of the guesses + 1 (ties broken randomly). Which of the following is an equilibrium: a) All announce 1. b) All announce 50. c) All announce 75. d) All announce 100(ii) A mixed strategy profile (p, q) is one in which p = (p,P2.... P) is the mixed strategy of player 1, and q- (g1, q2,..q4) is the mixed strategy of player 2. Show that if p, >0 in a Nash equilibrium profile (p*, q*), the player 2 must also play i with strictly positive probability q'; > 0. (State clearly any theorem you use to show this. You are not required to justify the theorem.) %3D
- The promoter of a football game is concerned that it will rain. She has the option of spending $14,040 on insurance that will pay $39,000 if it rains. She estimates that the revenue from the game will be $65,040 if it does not rain and $30,040 if it does rain. What must the chance of rain be if buying the policy has the same expected return as not buying it? Write expressions showing the expected returns if the promoter does and does not purchase the insurance, using p to represent the probability of rain. Without insurance, E(return) = With insurance, E(return) = The chance of rain must be _%.Mete is Esra's boyfriend. Esra is expecting a marriage proposal from Mete. Today is Sunday. Mete says Esra that: (1) he is going to propose Esra on Monday or Tuesday or Wednesday or Thursday or Friday, at 10:00 pm. (2) he knows right now the day when he will propose (3) but the day of proposal will be a surprise to Esra: On the day of the proposal, she would not be expecting the proposal that day. Esra says Mete that what he says is impossible. Explain why Esra is right in a few sentences.3. The utility function is u(x, y) = √x+y. Suppose that 1) with probability 0.5, (I, Px, Py) (8, 2, 2) and 2) with probability 0.5, (I, Px, Py) = (8, 4, 8). Explain graphically whether an increase in x increases or decreases the riskiness of the utility gamble. U 6 5 4 3 2 1 0 0 0.25 0.5 0.75 1 1 1.25 1.5 1.75 28
- 2. The utility function is u(x, y) = min{x,y}. Suppose that 1) with probability 0.5, (I, Px, Py) = (4, 1, 1) and 2) with probability 0.5, (I, Px, Py) (12, 6, 2). Explain graphically whether an increase in x increases or decreases the riskiness of the utility gamble. Y 6 5 4 3 2 1 0 0 1 2 IN I 3 4 5 = 6 XIn a final round of a MegaMillion TV show, a contestant has won $1 millionand has a chance of doubling the reward. If he loses his winnings drop to$500,000. The contestant thinks his chances of winning are 50%. Should heplay? What is the lowest probability of a correct guess that will make his betprofitable? Show work5.Each of Player 1 and Player 2 chooses an integer from the set {1, 2, ..., K}. If they choose the same integer, P1 gets +1 and P2 gets -1; if they choose different integers, P1 gets -1 and P2 gets +1. (a) Show that it is a NE for each player to choose every integer in {1, 2, ..., K} with equal probability, K1 . (b) Show that there are no NE besides the one you found in (a).