In a Roulette game, a ball falls onto a spinning wheel and randomly ends up in one of 37 pockets. The Casino claims that each pocket has the same probability to catch the ball and clients can bet on numbers. A client suspects that the Roulette table is rigged in favor of the number 7. She decides to observe the Roulette table for a couple of hours and counts the times the ball lands in the number-7 pocket. In this time she can observe 400 instances where the Roulette wheel is spun. a) State the explicit equation for the probability distribution for the number of times the ball hits the number-7 pocket, in the case that the Roulette is not rigged. Give all key parameters for this distribution and Calculate the expected number of times the ball hits the number-7 pocket and the standard deviation of this number. Also State the null-hypothesis H0 and the alternate hypothesis H1 in the context of the client’s claim. b). The Casino will only accept a client’s claim if the error probability is lower than 0.1%. Calculate the number of times the client has to observe the outcome “7” for her claim to be acepted by the casino. c). The player observes 9 instances where the ball ends up in the number-7 pocket. Explain whether her claim would be accepted by the Casino. Based on the client’s observation, state the 95% CL upper limit for the probability for the number-7 pocket. For this purpose, approximate the probability distribution by another probability distribution which allows you to use tabled values from the lecture notes. Justify why you can use this approximation. State the explicit calculation which results in the tabled value.
In a Roulette game, a ball falls onto a spinning wheel and randomly ends up in one of 37 pockets. The Casino claims that each pocket has the same
a) State the explicit equation for the probability distribution for the number of times the ball hits the number-7 pocket, in the case that the Roulette is not rigged. Give all key parameters for this distribution and Calculate the expected number of times the ball hits the number-7 pocket and the standard deviation of this number. Also State the null-hypothesis H0 and the alternate hypothesis H1 in the context of the client’s claim.
b). The Casino will only accept a client’s claim if the error probability is lower than 0.1%. Calculate the number of times the client has to observe the outcome “7” for her claim to be acepted by the casino.
c). The player observes 9 instances where the ball ends up in the number-7 pocket. Explain whether her claim would be accepted by the Casino. Based on the client’s observation, state the 95% CL upper limit for the probability for the number-7 pocket. For this purpose, approximate the probability distribution by another probability distribution which allows you to use tabled values from the lecture notes. Justify why you can use this approximation. State the explicit calculation which results in the tabled value.
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