11 Bob has glued himself to a certain slot machine for four hours in a row now with his bucket of coins and a bad attitude. He doesn't want to leave because he feels the longer he plays, the better chance he has t win eventually. Is poor Bob right?
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- At roulette, there are 38 slots. A "column" consists of 12 of these slots. A column pays 2 to 1. So, if you risk $1 on a column and win, you net gain +2 dollars. If you lose, your net equals –1 dollar. Suppose you will play for 100 rounds and bet one dollar on a column each round. (You may enter values rounded to the nearest cent. Beware of carried rounding error.) Your average net gain per play equals 0.0526 dollars. The SD of your net gain per play equals 1.39 dollars. The expected value of your total net gain after 100 plays equals -5.3 dollars. The standard error of your total net gain after 100 plays equals 13.94 dollars.If a person rolls doubles when tossing two dice, the roller profits $ 65 , that is, the roller receives $ 65 more than the cost of the game. If the game is fair, how much should the person pay to play the game?Fifty-eight out of 200 students are 12 or 13 years. Half of the rest are older than 13years and the other half are younger than 12 years. What percent of the students are 12 and 13? What percent of the students are older than 13? How many students are younger than 12? the fish market is selling several different kinds of fish. There are no prices listed. You ask about the prices, but the fish seller will only tell you this:• A pound of salmon and a pound of bass are $12.• A pound of bass and a pound of swordfish are $10.• A pound of salmon and a pound of swordfish are $8.• A pound of swordfish and a pound of catfish are $5.How much is each pound of fish? Show your work. (Each price per pound is a whole dollar amount.) A rectangular polygon consists of 8 squares placed side by side. If the perimeter of the rectangle is 180 inches, what is the area of the rectangle in square inches? A well is 24 feet deep and a snail climbs up 4 feet…
- Suppose you are playing a simple game that costs you $5 per play, where you draw one cardfrom a well-shuffled deck of 52 cards.• If the card is a spade, you receive $20. (You don’t get the $5 back.)• If it is not, then you do not receive any money. (You don’t get the $5 back.)Let X = the amount of money you have before playing a round of this “game”.Let Y = the amount of money you have after playing a round of this “game”.(a) Compute the conditional expectation of Y given X = $100.(b) You can also use the following variant of the Law of Total Expectation tocompute a conditional expectation:E(Z | X = x) = Summation(E(Z | Ai ∩ X = x)P (Ai | X = x)For this part, let Z = the amount of money you have after playing TWO rounds ofthis “game”. Use the above formula to calculate the conditional expectation of Z givenX = $100. (Note: the cards are well-shuffled again between rounds.)(c) Now suppose you get only $16 for drawing a spade. Compute the conditionalexpectation of Y (not Z) given X =…4. On a game show, you start with a prize pool of $1 000 000, but before you get it, you must answer trivia questions. Every time you get a question wrong, the current prize pool is cut in half, and you must keep answering questions. The game ends when you get a question right and you receive whatever is left of the prize pool. (For example, if you get the first two questions wrong, the prize pool is cut in half twice, so it becomes $250 000. If you then get the third question right, you go home with $250 000 as your winnings.) (a) If you have a chance of getting each question right, what is the distribution of Q: the total number of questions you have to answer? (Give the name of the distribution and its parameters.) (b) Express the amount of money you win as a function of Q. (c) Find the expected amount of money you win.Suppose I flip a fair coin n = 20 times. A psychic tries to predict the outcome before each flip. Three researchers have different ideas about the psychic's ability. There is Sydney, the Skeptic (S), who thinks the psychic's success rate is between 49% and 51%. There is Morgan, the Mark, M, who thinks that the psychic's success rate is 80%. And there is Carter, the Cynic (C), who thinks the psychic's success rate is 10%. Specifically: S+ 0 ~ U(.49, .51) M + 0 = .80 C0 = .10 %3D In all cases, assume the number of successful predictions follows a binomial distribution with success rate 0. Usek for the number of successes and n for the number of trials. Given all that: Determine the formula for the Bayes factor (a.k.a., likelihood ratio) supporting Carter over Morgan. Call that Bayes factor Bc:M Determine the formula for the Bayes factor (a.k.a., likelihood ratio) supporting Morgan over Sydney. Call that Bayes factor BM:S Determine the formula for the Bayes factor (a.k.a., likelihood…
- Mark is a university student and currently studying Statistics. He likes this girl, Lucy, and wants to ask her out on a date. He knows that he might be rejected, accepted, or getting a maybe. So he thought of asking her out twice instead of just once, in case he got rejected in the first try. He defines a success as getting at least one acceptance, no matter whether it is in the first or second try (or both if he's lucky!). What is the probability that he's lucky assuming that he's successful? (Round your answer to two decimal places.)Mike's game consists of selecting two balls, without replacement, from an urn containing 7 balls numbered 1 through 7. The amount that you win (in dollars) is the maximum of the numbers on the two balls. How much should be charged to play this game if the Mike wants to make $0.90, on average, each time the game is played?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) $250
- Janus Seagull had a car accident and was out of work for a year. Janus believes that the accident was caused by a vehicle defect. He consulted some lawyers and planned to sue the vehicle manufacturer. During negotiations, the legal team of the vehicle manufacturer offered a $700K settlement. However, Janus needs to settle the $100K in legal fees. Janus asked his lawyer for advice and the lawyer told him that he has a 50% chance of winning the case. If Janus loses, he will incur legal fees of $75K. If he wins, the full requested settlement is also not guaranteed. The lawyer believes that there is a 50% chance that Janus will receive full settlement of $3 million, of which Janus needs to settle $1 million in legal fees. There is a 50% chance that the jury will award Janus $1 million, of which 50% will be taken up by legal fees.3. Tamir has recently begun a new job of selling custom bracelets! Tamir's sales have been growing every week and now he is looking to analyze his profit. Specifically, Tamir is looking to find how much his cost of goods is for his business of selling bracelets. Luckily, Tamir sells each bracelet for the same price and has recorded that when he sold 4 bracelets he made a $39 profit, when he sold 9 bracelets he made a $94 profit. a. Write an equation in point slope form that could model this scenario. b. Which of the following could be the graph of Tamir's profit? Explain how you know. Graph 1: Graph 2: + +Bob and Doug play a lot of Ping-Pong, but Doug is a much better player, and wins 60% of their games.To make up for this, if Doug wins a game he will spot Bob five points in their next game. If Doug wins again he will spot Bob ten points the next game, and if he still wins the next game he will spot him fifteen points, and continue to spot him fifteen points as long as he keeps winning. Whenever Bob wins a game he goes back to playing the next game with no advantage.It turns out that with a five-point advantage Bob wins 60% of the time; he wins 70% of the time with a ten-point advantage and 90% of the time with a fifteen-point advantage.Model this situation as a Markov chain using the number of consecutive games won by Doug as the states. There should be four states representing zero, one, two, and three or more consecutive games won by Doug. Find the transition matrix of this system, the steady-state vector for the system, and determine the proportion of games that Doug will win in the…





