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Kai wants to create a fair dice game for two people, using divisibility by \(3 .\) He decided on these rules: \(\cdot\) Each player rolls one die. \(\cdot\) The players find the sum of the two numbers. \(\cdot\) Player 1 scores if the sum is divisible by \(3 .\) Player 2 scores if the sum is not divisible by \(3 .\) Kai isn't sure how many points each player should score each time. a. Make a table showing the probability of each sum being rolled. b. What is the probability that Player 1 scores on a given roll? c. Kai decided to give 2 points to Player 2 when the sum isn't divisible by \(3 .\) For the game to be fair, how many points should Player 1 score each time the sum is divisible by 3\(?\)

Step by step solution

01

- List Possible Outcomes

Create a table listing all possible outcomes of rolling two dice. A die has 6 faces, so there are 6x6=36 possible outcomes.

02

- Calculate Sum for Each Outcome

Find the sum for each possible outcome. For example, rolling a 1 and a 1 gives a sum of 2, rolling a 1 and a 2 gives a sum of 3, up to rolling a 6 and a 6 which gives a sum of 12.

03

- Count Each Possible Sum

Count how many times each sum can appear: 2 through 12. For example, sum 2 occurs once (1+1), sum 3 occurs twice (1+2 and 2+1), and so on.

04

- Make the Probability Table

Represent the frequency of each sum (from 2 to 12) in a probability table. For example, the probability of rolling a 7 is 6 out of 36 since there are 6 ways to get a 7.

05

- Calculate Probability for Sums Divisible by 3

Identify sums that are divisible by 3 and calculate their combined probability. Sums that are divisible by 3 are 3, 6, 9, and 12. Add the probabilities of these outcomes.

06

- Calculate Probability for Sums Not Divisible by 3

Identify sums that are not divisible by 3 and calculate their combined probability (i.e., the remaining sums).

07

- Set Up Equation for Fair Points Distribution

Use the probabilities to set up an equation ensuring expected values for scoring are equal for both players. Let Player 1’s points be X. Solve for X using the equations: Probability of Player 1 scoring * X = Probability of Player 2 scoring * 2.

08

- Solve for Player 1’s Points

Solve the equation from step 7 to find how many points Player 1 should score for a fair game.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Divisibility

In probability and number theory, **divisibility** is a fundamental concept. It's all about finding out if one number can be divided by another without leaving a remainder. For Kai's dice game, we need to check if the sum of two dice is divisible by 3. If a number is divisible by 3, it means when you divide that number by 3, there is no remainder.

For example, if the sum of the dice is 9, since 9 divided by 3 equals 3 with no remainder, 9 is divisible by 3.

This concept is crucial because Player 1 scores points when the sum of the dice roll is divisible by 3.

Sum of Two Dice

When rolling two dice, you can get sums ranging from 2 to 12. To visualize this, imagine one die shows a number between 1 and 6, and the other die also shows a number between 1 and 6. The smallest sum (1+1) equals 2, while the largest sum (6+6) equals 12.

Here are some examples:

• If you roll a 2 and a 5, the sum is 2 + 5 = 7.
• If you roll a 3 and a 3, the sum is 3 + 3 = 6.

When creating a probability table for this game, you list out all possible pairs of dice rolls and their sums.

Fair Game Calculation

To ensure Kai's game is fair, we calculate how many points each player should earn. A fair game means both players have an equal chance of winning. In other words, the expected values for both players should be equal.

First, we calculate probabilities:

• The probability that a sum is divisible by 3 (Player 1 scores) is found by adding the probabilities of sums that are multiples of 3.
• The probability that a sum is not divisible by 3 (Player 2 scores) is the remaining probability.

Let X be the points for Player 1. The game is fair if: \[ P(divisible) \times X = P(not divisible) \times 2 \] where P(divisible) and P(not divisible) are the probabilities for each player's outcomes.

Probability Table

A **probability table** helps visualize the outcomes and their probabilities. Here, we provide probabilities for each possible sum when rolling two dice. There are 36 possible outcomes (6 sides on the first die times 6 sides on the second die).

For instance:

• The sum of 2 can occur in only one way: (1, 1). Thus, the probability is 1/36.
• The sum of 7 can occur in six ways: (1, 6), (2, 5), (3, 4), (4,3), (5, 2), (6, 1). Thus, the probability is 6/36 or 1/6.

By filling out the table for all sums from 2 to 12, we can easily see which sums are more likely and calculate the overall probabilities needed for a fair game.