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Problem 1: The Suspicious Card GameAssigned: 12 OctoberDue: 28 OctoberMaximum Mark: 10Maximum Submission Length: 4 pagesYou’re playing a simple game of chance against a stranger. In each round of the game,each player draws two cards from what appears to be a standard playing-card deck (whichnormally contains 52 cards, four each from the set [A,2,3,4,5,6,7,8,9,10,J,Q,K]). The playerwith the highest card in their hand wins, and the losing player must pay them £10. (If theplayers have the same high card, no money changes hands that round.) Each player drawsnew cards each round, returning cards from the previous round to the deck. Aces are low.After the first five rounds you obtain the following results:

Round 1: You = [10][2]Round 2: You = [9][6]Round 3: You = [2][K]Round 4: You = [Q][1]Round 5: You = [7][2]

Opponent = [K][5]Opponent = [5][K]Opponent = [K][9]Opponent = [K][7]

(You LOSE)(You LOSE)(You DRAW)(You LOSE)

Opponent = [10][10] (You LOSE)

Without a win to your name, you begin to suspect that your opponent may be cheating.1. Develop and describe a mathematical model for the game that enables you to addressyour suspicions using statistical inference, and use it to argue for or against your suspicion.Specifically, do the following:(a) Define a test statistic: a value that can be calculated for any potential outcome ofthe game that encapsulates how unusual (or not) the result is. Determine the value of thetest statistic for the result that actually occurred, using the data at hand.(b) Define a null hypothesis: a scenario, expressed mathematically, that describes aspecific uninteresting configuration of the system. Also define an alternative hypothesis thatdescribes the scenario or scenarios that would be considered of interest.(c) Calculate the probability that a result (test statistic) “as or more extreme” than theactual result would result from the scenario under the null hypothesis. You may use ananalytic method, or a simulation.(d) Based on the previous calculation, draw a conclusion about the null hypothesisand/or alternative hypothesis.2. In practice, did the procedure you carried out follow all rules of statistical hypothesistesting, such that it constitutes a formal and unbiased means of evaluating whether youropponent might be cheating? Describe any important deviations from the assumptions ofhypothesis-testing, if present. (Note: this is a question about the procedure you followed, notabout the quantity or quality of data.)3. What steps could you take next to make your conclusion more secure?

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