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. b) Calculate the expected number of times the ball hits the number-7 pocket and the standard deviation of this number. c) State the null-hypothesis Ho and the alternate hypothesis H₁ in the context of the client's claim. d) The Casino will only accept a client's claim if the error probability is lower than 0.1%. Cal- culate the number of times the client has to observe the outcome "7" for her claim to be acepted by the casino. e) 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. f) 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 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. b) Calculate the expected number of times the ball hits the number-7 pocket and the standard deviation of this number. c) State the null-hypothesis Ho and the alternate hypothesis H₁ in the context of the client's claim. d) The Casino will only accept a client's claim if the error probability is lower than 0.1%. Cal- culate the number of times the client has to observe the outcome "7" for her claim to be acepted by the casino. e) 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. f) 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.
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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Transcribed Image Text: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.
b) Calculate the expected number of times the ball hits the number-7 pocket and the standard
deviation of this number.
c) State the null-hypothesis Ho and the alternate hypothesis H₁ in the context of the client's
claim.
d) The Casino will only accept a client's claim if the error probability is lower than 0.1%. Cal-
culate the number of times the client has to observe the outcome "7" for her claim to be
acepted by the casino.
e) 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.
f)
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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VIEWStep 2: State the explicit equation for the probability distribution for the number of times the ball hits 7
VIEWStep 3: Calculate the expected number of times the ball hits the number-7 pocket and the standard deviation
VIEWStep 4: State the null-hypothesis H0 and the alternate hypothesis H1 in this context
VIEWStep 5: Calculate the number of times the client has to observe the outcome "7" for her claim to be accepted
VIEWStep 6: Explain whether the given claim would be accepted by the Casino or not
VIEWStep 7: State the 95% CL upper limit for the probability for the number-7 pocket based on the given claim
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