Final answer:
The mean or expected value for a single roll of a fair six-sided die is 3.5. The variance is calculated as approximately 2.92, and the standard deviation as approximately 1.71, all of which summarize the distribution of outcomes for rolling the die.
Step-by-step explanation:
When you roll a fair six-sided die, the outcomes range from 1 to 6, each with an equal probability of 1/6. To calculate the mean (expected value), you add up all the possible outcomes and divide by the number of outcomes:
µ (mean) = (1 + 2 + 3 + 4 + 5 + 6) / 6 = 3.5
To calculate the variance, we use each outcome's deviation from the mean squared, multiplied by its probability, and sum all these products:
σ² (variance) = ∑ [(x - µ)² * P(x)] = (1/6)*((1-3.5)² + (2-3.5)² + (3-3.5)² + (4-3.5)² + (5-3.5)² + (6-3.5)²)
After performing the calculations, the variance σ² for a single die roll is found to be approximately 2.92.
The standard deviation is the square root of the variance:
σ (standard deviation) = √(σ²) ≈ √(2.92) ≈ 1.71
These statistical measures describe the distribution of outcomes for a single roll of a die.