Consider the following gambler's ruin problem. A gambler bets $1 on each play of a game. Each time, he has a probability p of winning and probability 1 - p of losing the dollar bet. He will continue to play until he goes broke or nets a fortune of T dollars. Let Xn denote the number of dollars possessed by the gambler after the nth play of the game. Then Xn+1 = = [X₂+1 (Xn - 1 Xn+1 = Xn with probability p with probability 1-p for 0 < X
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- You are bidding for an antique trombone in a sealed bid auction. All of your bids must be in increments of $10. You value the trombone at exactly $2,105, and guess that each time the price drops one $10 increment there is a 4% chance that someone else will bid (Assume that if you both bid then they other bidder gets it, as they bid a multiple of $10 plus $1..). The trombone goes to the highest bidder, the “winner” of the auction. Which of the following numbers represents a difference between the amount you should bid if the winner pays the amount they bid vs. if the winner pays an amount equal to the second-highest bid? a) $0 b) $80 c) $240 d) $250Suppose that Albert and Leo are in a contract dispute. Leo has breached their contract and this has resulted in damages of amount x to Albert. They can negotiate a settlement before they go to trial. If they settle, they divide the gain in total surplus from settling equally. Going to court costs each 1000 (and they are in the "American" system). Suppose that Albert anticipates that the court will find x to be 0 with probability and 20,000 with probability 4. Further, suppose that Leo anticipates that the court will find x to be O with probability ½ and 20,000 with probability %. Assume both Albert and Leo are risk neutral and motivated by their expected payoffs. Part A: Will Albert and Leo settle? If so, what is the outcome? If not, explain why. Part B: Suppose now that Albert believes the court will find x to be 0 with probability 1. Leo anticipates that the court will find x to be 0 with probability % and 20,000 with probability %. If Albert anticipates there is scope for settlement…a. In CSE230, there were 2 assignments this semester. The probability that you would NOT get above average marks in the first and the second assignments are 0.3 and 0.4 respectively. But if someone does NOT get the above average marks in the first assignment, the probability of NOT getting above average marks in the second assignment becomes 0.6. What is the probability of NOT getting above average marks in both the assignments? b. In a rational world, husband and wife are arguing over a matter. The probability of the husband being Wrong is 0.15 and the probability of the wife being Wrong is 0.13. And the probability of husband being Wrong provided that wife is Wrong is 0.18. What is the probability that the wife is Wrong provided that the husband is Wrong?
- a. In CSE230, there were 2 assignments this semester. The probability that you would NOT get above average marks in the first and the second assignments are 0.3 and 0.4 respectively. If someone gets the above average marks in the first assignment, the probability of getting above average marks in the second assignment becomes 0.9. What is the probability to get above average marks in both the assignments? b. In a rational world, husband and wife are arguing over a matter. The probability of the husband being Wrong is 0.2 and the probability of the wife being Wrong is 0.22. And the probability of husband being Wrong provided that wife is Wrong is 0.17. What is the probability that the wife is Wrong provided that the husband is Wrong?The graph shows that U.S. smokers have a greater probability of suffering from some ailments than the general adult population. The exercise is based on some of the probabilities, expressed as decimals, shown to the right of the bars. If the probability an event will occur is p and the probability it will not occur is q, then each term in the expansion of (p + q)n represents a probability. Solve, The probability that a smoker suffers from depression is 0.28. If five smokers are randomly selected, the probability that three of them will suffer from depression is the third term of the binomial expansion of What is this probability? Use a calculator to determine the probability, correct to four decimal places.A casino offers the following game, which costs 10 dollars to play. Two cards are selected at random without replacement from a standard deck of 52 cards. If both of the cards are kings, you win 220 dollars and get your 10 dollars back. If exactly one of the cards is a king, you win 20 dollars and get your 10 dollars back, Otherwise, you lose the 10 dollars. Let X be equal to the amount of money won as the result of playing this game one time, where winning a negative amount is equivalent to losing that amount. Find P(X>0) Find E(X)
- Find the value of x if the events A and B are independentA roulette wheel has 30 slots, consisting of 2 blues, 8 whites, and 20 red slots. You will receive P100 if the roulette stops spinning on a blue slot, you will receive P50 on a white slot, but you will be penalized P10 if the roulette stops spinning on a red slot. Let X be the amount you will recieve (or pay). The probaility of receiving P50 is 2/30 or 1/15, and of receiving P50 is 8/30 or 4/15. What is the probability of penalty of P10? 2/3 1/3 -2/32.)A jar contains 2 red, 3 green, and 6 blue marbles. In a game a player closes their eyes, reaches into the jar and randomly chooses two marbles. The player wins the game if at least one of their marbles is red. Suppose it costs $1 to play the game and the winning prize is $3. Mathematically analyze this game and determine if it is in your financial interest to play the game.
- Suppose you play a game that you can only either win or lose. The probability that you win any game is 65%, and the probability that you lose is 35%. Each game you play is independent. Let X is the number of wins If you play the game 20 times. P(X=13) = 20C13(Assume that the group has a portfolio of 6 stocks. There is 30% chance that any one of these stocks will increase in value. Find the proability that four of the six stocks increase in value.Suppose that an electronics store runs a promotion. At the cash register, each separate item is independently given an instant discount. The discount amounts and chances are: Discount 25% off 10% off None Probability 0.05 0.35 0.60 This means two items might get the same discount or they could get different discounts. I will purchase a camera priced at $200 and a video game priced at $80. Let X be the total amount saved in dollars at the cash register. It is possible that we wont save any money at all. What is the chance of this? P(X = 0 ). Express your answer as a proportion using 2 decimal places.