Problem 58 - Algebra and Trigonometry with Analytic Geometry

Question

Crapless craps Refer to Exercise 57. In the game of craps, a player loses a pass line bet if a sum of 2,3 , or 12 is obtained on the first roll (referred to as "craps"). In another version of the game, called crapless craps, the player does not lose by rolling craps and does not win by rolling an 11 on the first roll. Instead, the player wins if the first roll is a 7 or if the point \((2-12\), excluding 7\()\) is repeated before a 7 is rolled. Find the probability that the player wins on a pass line bet in crapless craps.

Step-by-Step Solution

Answer

The probability that the player wins in crapless craps is about 38.36%.

Step 1: Understanding the Dice Outcomes

In a game of craps, there are 36 possible outcomes when rolling two six-sided dice. Each die has outcomes from 1 to 6, so the combinations that sum to any given number can be determined. For this problem, consider the specific meaningful sums for crapless craps: 7 or repeating any of 2, 3, 4, 5, 6, 8, 9, 10, 11, or 12 before a 7 is rolled again.

Step 2: Determining Winning Outcomes on First Roll

In crapless craps, a player wins immediately if a 7 is rolled on the first roll. The number 7 can be rolled in 6 ways (1+6, 2+5, 3+4, 4+3, 5+2, 6+1). Thus, the probability of rolling a 7 on the first roll is \(\frac{6}{36} = \frac{1}{6}\).

Step 3: Understanding Points in Crapless Craps

If a player does not roll a 7 on the first roll, any other number becomes the 'point' except for 11 and 7. If a point is established (2, 3, 4, 5, 6, 8, 9, 10, or 12), the goal is to roll that number again before rolling a 7 to win.

Step 4: Probability of Winning with Each Point

The probability of winning with a specific point can be calculated as the ratio of ways to roll that point again versus rolling a 7. For example, if 4 is a point, we win if another 4 is rolled before a 7. Since there are 3 combinations for 4 and 6 for 7, the probability of winning if 4 is the point is \(\frac{3}{9} = \frac{1}{3}\). Repeat this for all other points.

Step 5: Calculating Total Win Probability

To find the total probability of winning from any scenario, add the probability of winning on the first roll to the probabilities of winning from any other point. Consider the probability of establishing a point and then successfully rolling that point again. Sum these probabilities to get the total win probability.

Step 6: Combining Probabilities

The total winning probability is the sum of probabilities from Step 2 and Step 4 (adjusted for the initial probability of each point being rolled): \(\frac{1}{6} + \sum \left(\text{Probability of Point} \times \text{Probability of Winning with Point}\right)\). Calculate for each point.

Key Concepts

Crapless Craps Rules

Crapless craps is an intriguing variant of the traditional craps game.
Unlike standard craps, you don't lose a pass line bet by rolling a "craps number" (2, 3, or 12) on your first roll. Similarly, you won’t win by rolling an 11 right away either.
Here's a breakdown of the game mechanics for crapless craps:

• If you roll a 7 on your first roll, you win immediately.
• If you roll any number from 2 to 6 or 8 to 12, that number becomes your 'point'.
• To win, your point must be repeated before you roll a 7 again.

This twist in rules changes the dynamics and probabilities involved, making it a distinctive and somewhat less predictable version of craps.

Dice Outcomes

When playing dice games like craps (or crapless craps), understanding dice outcomes is crucial.
Essentially, there are 36 possible outcomes when rolling two six-sided dice. This stems from each die having outcomes from 1 to 6, with various combinations leading to sums between 2 and 12.
For example:

• A sum of 7 can be achieved in 6 different ways: (1+6), (2+5), (3+4), (4+3), (5+2), and (6+1).
• Some numbers, like 2 (1+1) or 12 (6+6), only have one way to occur.

Knowing these combinations helps in calculating the probability of rolling certain numbers or achieving specific outcomes during the game.

Pass Line Bet Odds

In craps, and consequently in crapless craps, the odds of winning a pass line bet depend significantly on the outcomes of the dice rolls.
For crapless craps, if you're betting on the pass line:

• Your chances of winning immediately when rolling a 7 on the first roll are 1 in 6, due to the 6 ways to roll a 7 out of 36 possible outcomes.
• For any other point established, your odds vary based on how many combinations can make your point versus how many lead to a 7.

This variability in odds makes probability insights essential for strategic gameplay.

Probability Calculation

Calculating probabilities in crapless craps involves a series of straightforward yet detailed steps.
For the first roll:

• The probability of rolling a 7 is \(\frac{6}{36} = \frac{1}{6}\).
• If a point is rolled (any number from 2 to 6 or 8 to 12), you must calculate the probability of rolling that point again before a 7, based on its specific combinations.

To find the total probability of winning at crapless craps, sum the immediate win chance from rolling a 7 and the probabilities of winning with each point established.
Put simply, the formula is: \[\text{Total Win Probability} = \frac{1}{6} + \sum \left( \text{Probability of Point} \times \text{Probability of Winning with Point} \right)\]Understanding this framework helps players gauge their winning likelihood and adjust their strategies accordingly.