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Probability Your friend has a game night every week. For fun, he uses a fake 6-sided die on certain nights with probability P(D = f), otherwise, he uses the real 6-sided die. In this question, D = f denotes a fake die and D=-f denotes a real die. Re {1, 2, 3, 4, 5, 6} denotes the result of a roll. For example R= 1 means that a 1 was rolled. (a) Suppose you know the probability of rolling a 1, 2, 3, 4, and 5 given that the die is fake. In other words, you know P(R = 1|D = f), P(R= 2|D = f), P(R= 3|D= f), P(R= 4|D = f), and P(R= 5|D = f). Given this information, how would you find P(R= 6|D= f)? P(R = 6|D = f) = (b) This night, you are the first to roll. What is the probability that you roll a 6 (i.e. P(R = 6))? Use the product rule and marginalization to give your answer in terms of P(R = 6|D = f), P(R = 6|D = -).P(D = f), and P(D =-f). P(R = 6) = (c) Say you rolled a 6. The next turn, your friend rolls a 1. Given that you rolled a 6 and your friend rolled a 1, what is the probability of the die being fake (i.e. P(D= f\R1 =6, R2 = 1))? Use Baye's Rule to give your answer in terms of P(R1 = 6, R2 = 1|D = f), P(D= f), and P(R1 = 6, R2 = 1). %3D P(D = f|R1 = 6, R2 = 1) =

Probability Your friend has a game night every week. For fun, he uses a fake 6-sided die on certain nights with probability P(D = f), otherwise, he uses the real 6-sided die. In this question, D = f denotes a fake die and D=-f denotes a real die. Re {1, 2, 3, 4, 5, 6} denotes the result of a roll. For example R= 1 means that a 1 was rolled. (a) Suppose you know the probability of rolling a 1, 2, 3, 4, and 5 given that the die is fake. In other words, you know P(R = 1|D = f), P(R= 2|D = f), P(R= 3|D= f), P(R= 4|D = f), and P(R= 5|D = f). Given this information, how would you find P(R= 6|D= f)? P(R = 6|D = f) = (b) This night, you are the first to roll. What is the probability that you roll a 6 (i.e. P(R = 6))? Use the product rule and marginalization to give your answer in terms of P(R = 6|D = f), P(R = 6|D = -).P(D = f), and P(D =-f). P(R = 6) = (c) Say you rolled a 6. The next turn, your friend rolls a 1. Given that you rolled a 6 and your friend rolled a 1, what is the probability of the die being fake (i.e. P(D= f\R1 =6, R2 = 1))? Use Baye's Rule to give your answer in terms of P(R1 = 6, R2 = 1|D = f), P(D= f), and P(R1 = 6, R2 = 1). %3D P(D = f|R1 = 6, R2 = 1) =

A First Course in Probability (10th Edition)
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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Probability Your friend has a game night every week. For fun, he uses a fake 6-sided die on certain nights with probability P(D = f), otherwise, he uses the real 6-sided die. In this question, D = f denotes a fake die and D = -f denotes a real die. RE {1, 2, 3, 4, 5, 6} denotes the result of a roll. For example R = 1 means that a 1 was rolled. (a) Suppose you know the probability of rolling a 1, 2, 3, 4, and 5 given that the die is fake. In other words, you know P(R = 1|D = f), P(R= 2|D = f), P(R = 3|D = f), P(R = 4|D = f), and P(R = 5|D = f). Given this information, how would you find P(R = 6|D= f)? P(R = 6|D = f) = (b) This night, you are the first to roll. What is the probability that you roll a 6 (i.e. P(R = 6))? Use the product rule and marginalization to give your answer in terms of P(R = 6|D = f), P(R = 6|D = -f),P(D = f), and P(D=¬f). P(R = 6) = (c) Say you rolled a 6. The next turn, your friend rolls a 1. Given that you rolled a 6 and your friend rolled a 1, what is the probability of the die being fake (i.e. P(D = f\R1 = 6, R2 = 1))? Use Baye's Rule to give your answer in terms of P(R1 = 6, R2 = 1|D = f), P(D= f), and P(R1 = 6, R2 = 1). P(D = f|R1 = 6, R2 = 1) =
Transcribed Image Text:Probability Your friend has a game night every week. For fun, he uses a fake 6-sided die on certain nights with probability P(D = f), otherwise, he uses the real 6-sided die. In this question, D = f denotes a fake die and D = -f denotes a real die. RE {1, 2, 3, 4, 5, 6} denotes the result of a roll. For example R = 1 means that a 1 was rolled. (a) Suppose you know the probability of rolling a 1, 2, 3, 4, and 5 given that the die is fake. In other words, you know P(R = 1|D = f), P(R= 2|D = f), P(R = 3|D = f), P(R = 4|D = f), and P(R = 5|D = f). Given this information, how would you find P(R = 6|D= f)? P(R = 6|D = f) = (b) This night, you are the first to roll. What is the probability that you roll a 6 (i.e. P(R = 6))? Use the product rule and marginalization to give your answer in terms of P(R = 6|D = f), P(R = 6|D = -f),P(D = f), and P(D=¬f). P(R = 6) = (c) Say you rolled a 6. The next turn, your friend rolls a 1. Given that you rolled a 6 and your friend rolled a 1, what is the probability of the die being fake (i.e. P(D = f\R1 = 6, R2 = 1))? Use Baye's Rule to give your answer in terms of P(R1 = 6, R2 = 1|D = f), P(D= f), and P(R1 = 6, R2 = 1). P(D = f|R1 = 6, R2 = 1) =
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