Question 1191019: Toss a four-sided die twice. Role-playing games like Dungeons & Dragons use many different types of dice. Suppose that a four-sided die has faces marked 1, 2, 3, and 4. The intelligence of a character is determined by rolling this die twice and adding 1 to the sum of the spots. The faces are equally likely, and the two rolls are independent. What is the average (mean) intelligence for such characters? How spread out are their intelligences, as measured by the standard deviation of the distribution?
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Here's how to calculate the average intelligence and its standard deviation:
**1. Possible Outcomes and Probabilities:**
When rolling a four-sided die twice, there are 4 * 4 = 16 possible outcomes. Since each face is equally likely, and the rolls are independent, each outcome has a probability of 1/16.
**2. Sums and Frequencies:**
We need to find the sum of the two rolls for each outcome and then add 1 to get the intelligence score. Here's a table:
| Roll 1 | Roll 2 | Sum | Intelligence |
|---|---|---|---|
| 1 | 1 | 2 | 3 |
| 1 | 2 | 3 | 4 |
| 1 | 3 | 4 | 5 |
| 1 | 4 | 5 | 6 |
| 2 | 1 | 3 | 4 |
| 2 | 2 | 4 | 5 |
| 2 | 3 | 5 | 6 |
| 2 | 4 | 6 | 7 |
| 3 | 1 | 4 | 5 |
| 3 | 2 | 5 | 6 |
| 3 | 3 | 6 | 7 |
| 3 | 4 | 7 | 8 |
| 4 | 1 | 5 | 6 |
| 4 | 2 | 6 | 7 |
| 4 | 3 | 7 | 8 |
| 4 | 4 | 8 | 9 |
**3. Average (Mean) Intelligence:**
To find the mean, we can sum all the intelligence scores and divide by the number of outcomes (16). Or, since each outcome is equally likely, we can average the possible sums and then add 1.
Average Sum = (2+3+4+5+3+4+5+6+4+5+6+7+5+6+7+8) / 16 = 5
Average Intelligence = Average Sum + 1 = 5 + 1 = 6
**4. Standard Deviation of Intelligence:**
1. **Calculate the variance:**
Variance = Σ[(Intelligence - Mean Intelligence)² * Probability of that Intelligence]
Here's a more efficient way to calculate the variance, using the sums:
Average of Sums Squared = (2² + 3² + ... + 8²) / 16 = 30
Variance of Sums = 30 - 5² = 5
Variance of Intelligence = Variance of Sums = 5
2. **Calculate the standard deviation:**
Standard Deviation = √Variance = √5 ≈ 2.24
**Answer:**
The average intelligence is 6. The standard deviation of intelligence is approximately 2.24.
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