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Chance of winning at “craps.” A version of the dice game
Chapter 3, Problem 126SE(choose chapter or problem)
Problem 126SE
Chance of winning at “craps.” A version of the dice game “craps” is played in the following manner. A player starts by rolling two balanced dice. If the roll (the sum of the two numbers showing on the dice) results in a 7 or 11, the player wins. If the roll results in a 2 or a 3 (called craps), the player loses. For any other roll outcome, the player continues to throw the dice until the original roll outcome recurs (in which case the player wins) or until a 7 occurs (in which case the player loses).
a. What is the probability that a player wins the game on the first roll of the dice?
b. What is the probability that a player loses the game on the first roll of the dice?
c. If the player throws a total of 4 on the first roll, what is the probability that the game ends (win or lose) on the next roll?
Questions & Answers
QUESTION:
Problem 126SE
Chance of winning at “craps.” A version of the dice game “craps” is played in the following manner. A player starts by rolling two balanced dice. If the roll (the sum of the two numbers showing on the dice) results in a 7 or 11, the player wins. If the roll results in a 2 or a 3 (called craps), the player loses. For any other roll outcome, the player continues to throw the dice until the original roll outcome recurs (in which case the player wins) or until a 7 occurs (in which case the player loses).
a. What is the probability that a player wins the game on the first roll of the dice?
b. What is the probability that a player loses the game on the first roll of the dice?
c. If the player throws a total of 4 on the first roll, what is the probability that the game ends (win or lose) on the next roll?
ANSWER:
Solution:
Step 1 of 4:
Given that, a player starts by rolling two balanced dice. If the roll (the sum of the two numbers showing on the dice) results in a 7 or 11, the player wins.
There are a total of outcomes when rolling 2 dice. If we let the first number in the pair represent the outcome of die number 1 and the second number in the pair represent the outcome of die number 2, then the possible outcomes are
1,1 2,1 3,1 4,1 5,1 6,1
1,2 2,2 3,2 4,2 5,2 6,2
1,3 2,3 3,3 3,3 5,3 6,3
1,4 2,4 3,4 4,4 5,4 6,4
1,5 2,5 3,5 4,5 5,5 6,5
1,6 2,6 3,6 4,6 5,6 6,6
If both dies are fair, then each of these outcomes are equally like and have a probability of