RMcCormick_Module 04 Applying Game Theory_012824

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Module 04 Assignment Applying Game Theory Rhonda McCormick Rasmussen University G163/STA1625 Section 03 Essential Statistics and Analytics Instructor: Nigel Basta 01/28/24

Applying Game Theory Part One: Dice roll data table: x 1 2 3 4 5 6 Total P(x) 3/25 4/25 3/25 2/25 10/25 3/25 x•P(x) 3/25 8/25 9/25 8/25 50/25 18/25 96/25 x•P(x) Fraction 0.12 0.32 0.36 0.32 2 0.72 3.84 P(x) Fraction 0.12 0.16 0.12 0.08 0.4 0.12 1 Part Two: 1. What is the expected outcome for rolling a six-sided die using the discrete probability distribution table above? μ =∑[x•P(x)] x is the value of dice roll and P(x) is the corresponding probabilities of x. ∑[x] = 1(0.12) + 2(0.32) + 3(0.36) + 4(0.32) + 5(2) + 6(0.72) = 3.84 2. What is the probability of rolling an even number according to the discrete probability distribution table above? The even numbers are 2, 4, and 6. P(x=even) =P(x=2) or P(x=4) or P(x=6) P(x=2) = 0.16 P(x=4) = 0.08 P(x=6) = 0.12

P(x=even) = 0.16 + 0.08 + 0.12 = 0.36  How does this compare to the theoretical probability of 0.5? The experimental probability is 0.36 which is 0.14 lower than 0.5 theoretical theory.  Explain why you think there is a difference between the theoretical probability and the experimental probability you found. Probability can not be assumed or predicted. The experiment can have a different outcome every time. 3. Create a binomial probability distribution based on the discrete probability distribution table above where a success is rolling an even number. Answer the following questions:  How do you know this is a Binomial Probability Distribution? Explain by showing how this example fits all four properties of a Binomial Probability Distribution. With 25 trials and even being a pass, odd being a fail. There is a fixed number of trials. There are 2 possible outcomes with even or odd. The probability of success is the same for each trial. The outcome of one trial does not affect the next trial.  Define n,p,q. n: the total number of trials. (25) p: the probability of success or the probability of getting an even number. (0.36) q: the probability of failure or the probability of getting an odd number. (0.64)  What is the probability that you will roll exactly 12 even numbers? 12 is the number of successful trials (x) 25 is the number of throws (trials) 0.36 is the probability =  What is the probability that you will roll at least 12 even numbers? =

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 Find the expected number of even numbers that you will roll. = 9 Part Three: The time it takes to finish a game of Monopoly is normally distributed with a mean of 120 minutes and a standard deviation of 30 minutes. Using this premise, answer the following questions. Show all work for full credit. Calculations should be performed in Excel while answers including an explanation of steps using proper terminology are provided in a separate document. Explain why this is a continuous probability distribution instead of a discrete probability distribution. With 2 specific time points, there are infinite points in between those 2 points. What is the probability that a game lasts less than 45 minutes? Distribute = 45 Mean = 120 Standard deviation = 30 = 0.0062 What is the probability that a game lasts more than 160 minutes? Distribute = 160 Mean = 120 Standard deviation = 30 = 0.0912 What is the z-score of a game that lasts exactly 105 minutes? Z= (x-μ)/ σ (105-120)/30 = -0.5

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