In a game of chance a player throws a pair of dice and scores points equal to the difference between the numbers on the two dice. Winner is the person who scores exactly `5` points more than his opponent. If two players are playing this game only one time, then the probability that neither of them wins to

Options

A. `(1)/(54)`
B. `(1)/(108)`
C. `(53)/(54)`
D. `(107)/(108)`

Explanation

To solve the problem step by step, let's break down the process of determining the probability that neither player wins in the game of chance involving two players throwing a pair of dice.

### Step 1: Understand the Game Rules
In this game, each player rolls a pair of dice, and they score points equal to the absolute difference between the numbers on the two dice. A player wins if they score exactly 5 points more than their opponent.

### Step 2: Calculate Possible Outcomes for One Player
When rolling two dice, the possible outcomes for the difference between the two numbers can be calculated. The possible differences (D) can be:
- D = 0 (when both dice show the same number)
- D = 1 (e.g., (1,2), (2,1))
- D = 2 (e.g., (1,3), (3,1))
- D = 3 (e.g., (1,4), (4,1))
- D = 4 (e.g., (1,5), (5,1))
- D = 5 (e.g., (1,6), (6,1))

The only differences that can lead to a win for Player A or Player B are when one player scores 5 points more than the other. The only way to achieve a difference of 5 is if one player rolls a (1,6) or (6,1).

### Step 3: Count Winning Combinations
For Player A to win, they must roll a difference of 5, which can happen in 2 ways:
1. Player A rolls (1,6)
2. Player A rolls (6,1)

For Player B, they can roll any combination that results in a difference of 0, 1, 2, 3, or 4, but not 5. The total number of outcomes for Player B that do not result in a difference of 5 is calculated by considering all possible outcomes (36) minus the winning outcomes for Player A.

### Step 4: Total Winning Outcomes
The total number of outcomes where either Player A or Player B wins can be calculated as:
- Player A wins in 2 ways (as calculated above).
- Player B can win in similar ways, leading to a total of 4 winning combinations (2 for A and 2 for B).

### Step 5: Calculate Total Outcomes
The total number of outcomes when both players roll the dice is:
- Total outcomes = 6 (sides of die) × 6 (sides of die) = 36.

### Step 6: Calculate Outcomes Where Neither Wins
To find the outcomes where neither player wins, we can subtract the total winning outcomes from the total outcomes:
- Total outcomes where neither wins = Total outcomes - Winning outcomes = 36 - 4 = 32.

### Step 7: Calculate the Probability
The probability that neither player wins is given by the ratio of the number of outcomes where neither wins to the total outcomes:
- Probability = (Number of outcomes where neither wins) / (Total outcomes) = 32 / 36 = 8 / 9.

### Final Answer
Thus, the probability that neither player wins is **8/9**.

---