3. Two players, High and Low, play the following game on the "board" shownbelow. They place a coin on the top node, and then they take turns to move thecoin one node downward until the coin arrives at one of the sixteen terminalnodes. High's goal is that the game ends with the coin at a node with as high avalue as possible, while Low's aim is that the target node has as low a value aspossible. Assuming that both of them play optimally, what is the outcome (thevalue of the last node) of the game if:(a) High starts?(b) Low starts?A1 14 7 12 3 16 6 10 2 13 4 11 5 15 8 9

High wants to reach 16 and low wants to reach 1. Both need 4 steps to reach their aim.

Answered: 3. Two players, High and Low, play the…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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The game of Chomp is played by two players. In this game, cookies are laid out on a rectangular grid. The cookie in the top left position is poisoned. The two players take turns making moves; at each move, a player is required to eat a remaining cookie, together with all cookies to the right and/or below (that is all the remaining cookies in the rectangle, in which the first cookie eaten is the top left corner). The loser is the player who has no choice but to eat the poisoned cookie. Prove that if the board is square (and bigger than 1 × 1) then the first player has a winning strategy.
please do the last question(Q3) as well Define a game as follow: you begin with an urn that contains a mixture of black and white balls, and during the game you have access to as many extra black and white balls as you need. In each move of the game, you remove two balls from the urn without being able to see what colour they are. Then you look at their colour and do the following: If the balls are the same colour, you keep them out of the urn and put a black ball in the urn. if the balls are different colours, you keep the black one out of the urn and put the white one back into the urn. Each move reduces the number of balls by one, and the game will end when only one ball is left in the urn. In this you will figure out how to predict the colour of the last ball in the urn and prove your answer using mathematical induction. Q1) Draw diagrams to map out all the possibilities for playing the game starting with two balls in the urn, then three balls, then four balls. For each case…
2.)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.
In the "normal version" of a game called NIM, there are two players. At the start, two piles of matches are placed on the table in front of them, each containing two matches. In turn, the players take any (positive) number of matches from one of the piles. The player taking the last match wins. Assuming optimal play, which player is sure to win in this game? O The winner cannot be determined. O Player 1 O Player 2
Two players play a game with the following rules: Player 1 puts on a blindfold while player two throws a die. The number that he gets is put aside as the "Target value". Then player two throws another die three times while giving player 1 information on weather he gets a lesser number than the target, equal to the target or greater than the target. Imagine that you are player one and you are handed the information that player two got [Lesser, equal, greater]. Deduce the probabilities of each possible target given the information.
The game Capture is played by two players, First and Second, who take turns to move towards one another on a narrow bridge. Initially, the two players are at opposite ends of the bridge, at a distance of n feet from each other. (Here n is an arbitrary positive integer.) When it is their turn, they are allowed to jump 1, 2, or 3 feet toward each other. The game starts by First making a move first; the game is over when one player is able to capture (jump on top of) the other player. Generalize the problem further to the case when the initial distance is n feetand the players are allowed to jump any (positive) integer number of feet up to k feet. (Here n and k are arbitrary natural numbers.)
Two players, A and B, are playing an asymmetrical game. There are n points on the game board. Each turn player A targets a pair of points and player B says whether those two points are connected or unconnected. A can target each pair only once and the game ends when all pairs have been targeted. Player B wins iff a point is connected with all other points on the very last turn, while player A wins if any point is connected with all other points on any turn but the very last one OR if no point is connected to all other points after the last turn. For what values of n does either player have a winning strategy?
An archery target consists of a gold circle with radius of 5 inches surrounded by a red ring whose diameter is 20 inches. Next, there are three progressively larger rings – blue, black, and white. The diameter of each ring is 10 inches larger than the diameter of the previous ring. John is a beginner so his aim is poor and, if his arrow hits the target, its location on the target is completely at random. John shoots and then yells, “I hit it!” Without using your calculator, find the following probabilities. Write the answers as a percentage. a) The probability his arrow landed in the gold circle is ______. b) The probability his arrow landed in the white area is ______. c) The probability his arrow landed in the red or the blue area is ______.
A charity organization is selling $50 raffle tickets as part of a fund-raising program. The grand prize is a trip to Mexico valued at $12750, and the runner-up prize is a weekend spa package valued at $1500. Twenty other players will receive $250 gift cards for a local grocery store. The organization will sell a total of 4500 tickets, and the winners will be selected via random drawing. Let x = net winnings earned from buying one ticket for this raffle (i.e. x = ticket winnings - ticket cost) a) Write a valid probability distribution (in table format) for x. Make sure you consider ALL of the possible outcomes associated with buying a ticket. X P(x) b) Use your distribution to determine the EXPECTED VALUE associated with buying a single raffle ticket. Show valid supporting work! time you
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)
A game theorist is walking down the street in his neighborhood and finds $20. Just as he picks it up, two neighborhood kids, Jane and Tim, run up to him, asking if they can have it. Because game theorists are generous by nature, he says he’s willing to let them have the $20, but only according to the following procedure: Jane and Tim are each to submit a written request as to their share of the $20. Let t denote the amount that Tim requests for himself and j be the amount that Jane requests for herself. Tim and Jane must choose j and t from the interval [0,20]. If j + t ≤ 20, then the two receive what they requested, and the remainder, 20 - j - t, is split equally between them. If, however, j + t > 20, then they get nothing, and the game theorist keeps the $20. Tim and Jane are the players in this game. Assume that each of them has a payoff equal to the amount of money that he or she receives. Find all Nash equilibria.
At a county fair, you pay $2 to play the following game: You are given a fair coin, and two bags standin front of you, labelled Bag A and Bag B. You flip a coin: If your coin lands Heads, you pick a ballfrom Bag A. If your coin lands tails, you pick a ball from Bag B. Each ball has a dollar value written onit, and you win the number of dollars shown on the ball. • Bag A contains four (4) balls in total: Three (3) balls show $1, and one (1) ball shows $5.• Bag B contains three (3) balls in total: Two (2) balls show $2, and one (1) ball shows $3.You WIN money if you leave with more money than you started with. You LOSE money if you leavewith less money than you started with. You BREAK EVEN if you leave with the same money that youstarted with.Let H denote the event that you flip heads, T denote the event that you flip tails, W the event that youwin, and L the event that you lose.(i) You pay to play exactly one game. What is the probability that you WIN given thatyou flipped a Heads?(ii)…

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