when a die was rolled 20 times, 4 came up five times a.) determine the empirical probability of rolling a 4 with a die based on the 20 trials. b.) determine the theoretical probability of rolling a 4 with a die. c.) how can you account for the difference between the results of parts a.) and b.)?

### a) **Empirical Probability**
The empirical probability is calculated based on actual experimental data. Given that the die was rolled 20 times and a 4 came up 5 times, the empirical probability of rolling a 4 is:

\[
\text{Empirical Probability} = \frac{\text{Number of times 4 came up}}{\text{Total number of rolls}} = \frac{5}{20} = 0.25
\]

### b) **Theoretical Probability**
For a fair 6-sided die, the theoretical probability of rolling any specific number (including 4) is:

\[
\text{Theoretical Probability} = \frac{1}{6} \approx 0.1667
\]

### c) **Difference Between Empirical and Theoretical Probability**
The difference between the empirical probability (0.25) and the theoretical probability (0.1667) can be attributed to **random variation** or **sampling variability**. In a small number of trials (such as 20 rolls), it is common to observe some deviation from the theoretical probability. Over a much larger number of trials, the empirical probability would likely get closer to the theoretical value due to the law of large numbers, which states that as the number of trials increases, the empirical probability tends to converge to the theoretical probability.

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### Related Questions:
1. What is the law of large numbers in probability theory?
2. How does increasing the number of trials affect the empirical probability?
3. Can we ever achieve perfect alignment between empirical and theoretical probabilities in real-world experiments?
4. How are empirical probabilities useful in real-world scenarios?
5. How can we calculate variance in empirical results to compare with theoretical values?

**Tip:** Empirical probability often fluctuates more with fewer trials; larger datasets provide more accurate representations.

Explore how to calculate both empirical and theoretical probability by analyzing the results of rolling a die 20 times. Learn how empirical probability differs from theoretical probability and how random variation plays a role in small sample sizes. Suitable for students in Grades 6-8.

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