The Mean Value of Outcomes from a Fair Six-Face Dice Roll

The mean value of the outcomes from a fair six-face dice roll is indeed 3.5. To understand why this is the correct answer, let's break it down step by step:

1. Understanding the Mean
The mean, also known as the average, is a measure of central tendency that represents the typical value of a set of numbers. It is calculated by summing up all the numbers in the set and dividing the sum by the total number of values.

2. Possible Outcomes of a Fair Six-Face Dice Roll
A fair six-face dice has six equally likely outcomes, which are the numbers 1, 2, 3, 4, 5, and 6. Each outcome has a probability of 1/6 since there are six equally likely possibilities.

3. Calculating the Mean
To calculate the mean value of the outcomes from a fair six-face dice roll, we need to find the average of the possible outcomes. We can do this by summing up all the possible outcomes and dividing the sum by the total number of outcomes.

- The sum of all the possible outcomes is 1 + 2 + 3 + 4 + 5 + 6 = 21.
- The total number of outcomes is 6.

Therefore, the mean value is 21/6 = 3.5.

4. Understanding the Result
The mean value of 3.5 indicates that, on average, the outcomes from a large number of fair six-face dice rolls will tend to converge towards 3.5. This means that if we roll the dice many times and calculate the average of the outcomes, it will approach 3.5.

5. Fairness of the Dice
The fact that the dice is fair ensures that each outcome has an equal chance of occurring. In other words, the probability of rolling any specific number is the same for all the numbers: 1/6. This fairness ensures that the mean value is not skewed towards any particular outcome.

Conclusion
In summary, the mean value of the outcomes from a fair six-face dice roll is 3.5. This value represents the average of the equally likely outcomes and provides a measure of central tendency for the dice roll results. The fairness of the dice ensures that each outcome has an equal chance of occurring, resulting in a balanced mean value.