A favorite casino game of dice “craps” is played in the following manner: A player starts by rolling a pair of balanced dice. If the roll (the sum of two numbers showing on the dice) results in a 7 or 11, the player wins. If the roll results in a 2 or 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).

Question

A favorite casino game of dice “craps” is played in the following manner: A player starts by rolling a pair of balanced dice. If the roll (the sum of two numbers showing on the dice) results in a 7 or 11, the player wins. If the roll results in a 2 or 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). 
When answering the following questions, you can use this outcome chart for the roll of two dice: 
Provide the probability answers in fraction and in decimal forms rounded to 4 digits.
a.	List the possible outcomes (sample space) for winning on the first roll of the dice.

b.	What is the probability that a player wins the game on the first roll of the dice?

c.	List the possible outcomes (sample space) for losing on the first roll of the dice.

d.	What is the probability that a player loses the game on the first roll of the dice?

e.	If the player throws a total of 4 on the first roll, what is the probability that the player wins in the next roll?

f.	If the player throws a total of 6 on the first roll, what is the probability that the player loses in the next roll?

GPT-4o mini · Answer

To solve these questions related to the game of craps using a pair of balanced dice, let's start by determining the outcomes of rolling two dice.

The total number of outcomes when rolling two six-sided dice is \(6 \times 6 = 36\).

The possible sums from rolling two dice are from 2 to 12. We will first list the outcomes, then calculate the probabilities for winning, losing, and specific scenarios based on given initial rolls.

a. List the possible outcomes (sample space) for winning on the first roll of the dice.

Winning outcomes on the first roll are:

A roll of 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1)
A roll of 11: (5,6), (6,5)

So, the winning outcomes are:

7: 6 outcomes
11: 2 outcomes

Total outcomes for winning = 6 (for 7) + 2 (for 11) = 8 outcomes.

b. What is the probability that a player wins the game on the first roll of the dice?

The probability of winning on the first roll is calculated as: \[ P(\text{Win}) = \frac{\text{Number of winning outcomes}}{\text{Total outcomes}} = \frac{8}{36} = \frac{2}{9} \approx 0.2222 \]

c. List the possible outcomes (sample space) for losing on the first roll of the dice.

Losing outcomes on the first roll are:

A roll of 2: (1,1)
A roll of 3: (1,2), (2,1)

So, the losing outcomes are:

2: 1 outcome
3: 2 outcomes

Total outcomes for losing = 1 (for 2) + 2 (for 3) = 3 outcomes.

d. What is the probability that a player loses the game on the first roll of the dice?

The probability of losing on the first roll is calculated as: \[ P(\text{Lose}) = \frac{\text{Number of losing outcomes}}{\text{Total outcomes}} = \frac{3}{36} = \frac{1}{12} \approx 0.0833 \]

e. If the player throws a total of 4 on the first roll, what is the probability that the player wins in the next roll?

If the first roll is 4, the player needs to roll a 4 again to win or roll a 7 to lose.

Outcomes for rolling a 4 are:

(1,3), (2,2), (3,1) — 3 outcomes.

Outcomes for rolling a 7 are:

(1,6), (2,5), (3,4), (4,3), (5,2), (6,1) — 6 outcomes.

Total outcomes possible = Outcomes for winning + Outcomes for losing = 3 + 6 = 9.

The probability of winning (rolling a 4) in the next roll is: \[ P(\text{Win | first roll is 4}) = \frac{\text{Number of outcomes for winning (4)}}{\text{Total outcomes}} = \frac{3}{9} = \frac{1}{3} \approx 0.3333 \]

f. If the player throws a total of 6 on the first roll, what is the probability that the player loses in the next roll?

If the first roll is 6, the player needs to roll a 6 again to win or roll a 7 to lose.

Outcomes for rolling a 6 are:

(1,5), (2,4), (3,3), (4,2), (5,1) — 5 outcomes.

Outcomes for rolling a 7 are the same as before:

(1,6), (2,5), (3,4), (4,3), (5,2), (6,1) — 6 outcomes.

Total outcomes possible = Outcomes for winning + Outcomes for losing = 5 + 6 = 11.

The probability of losing (rolling a 7) in the next roll is: \[ P(\text{Lose | first roll is 6}) = \frac{\text{Number of outcomes for losing (7)}}{\text{Total outcomes}} = \frac{6}{11} \approx 0.5455 \]

Summary of Results:

a. Winning outcomes: \( {(1,6), (2,5), (3,4), (4,3), (5,2), (6,1), (5,6), (6,5)} \) – 8 outcomes
b. Probability of winning: \( \frac{2}{9} \approx 0.2222 \)
c. Losing outcomes: \( {(1,1), (1,2), (2,1)} \) – 3 outcomes
d. Probability of losing: \( \frac{1}{12} \approx 0.0833 \)
e. Probability of winning next roll (if first roll is 4): \( \frac{1}{3} \approx 0.3333 \)
f. Probability of losing next roll (if first roll is 6): \( \frac{6}{11} \approx 0.5455 \)