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Consider the following game: A player throws a fair die
Chapter 3, Problem 187SE(choose chapter or problem)
Problem 187SE
Consider the following game: A player throws a fair die repeatedly until he rolls a 2, 3, 4, 5, or 6. In other words, the player continues to throw the die as long as he rolls 1s. When he rolls a “non-1,” he stops.
a What is the probability that the player tosses the die exactly three times?
b What is the expected number of rolls needed to obtain the first non-1?
c If he rolls a non-1 on the first throw, the player is paid $1. Otherwise, the payoff is doubled for each 1 that the player rolls before rolling a non-1. Thus, the player is paid $2 if he rolls a 1 followed by a non-1; $4 if he rolls two 1s followed by a non-1; $8 if he rolls three 1s followed by a non-1; etc. In general, if we let Y be the number of throws needed to obtain the first non-1, then the player rolls (Y − 1) 1s before rolling his first non-1, and he is paid 2Y −1 dollars. What is the expected amount paid to the player?
Questions & Answers
QUESTION:
Problem 187SE
Consider the following game: A player throws a fair die repeatedly until he rolls a 2, 3, 4, 5, or 6. In other words, the player continues to throw the die as long as he rolls 1s. When he rolls a “non-1,” he stops.
a What is the probability that the player tosses the die exactly three times?
b What is the expected number of rolls needed to obtain the first non-1?
c If he rolls a non-1 on the first throw, the player is paid $1. Otherwise, the payoff is doubled for each 1 that the player rolls before rolling a non-1. Thus, the player is paid $2 if he rolls a 1 followed by a non-1; $4 if he rolls two 1s followed by a non-1; $8 if he rolls three 1s followed by a non-1; etc. In general, if we let Y be the number of throws needed to obtain the first non-1, then the player rolls (Y − 1) 1s before rolling his first non-1, and he is paid 2Y −1 dollars. What is the expected amount paid to the player?
ANSWER:
Solution:
Step 1 of 4:
It is given that a player rolls a die repeatedly until he rolls 2,3,4,5 ,6.
He rolls the die until he gets 1 and stops when he get non 1.
Using this information we need to find the required values.