Question

match the frequency distribution of 180 rolls of a dodecahedron (a 12-sided die) with one of the histograms shown below.

which histogram shows a distribution that would likely result from 180 rolls of a dodecahedron?

a.
b.
c.
d.

Explanation:

Analyze the theoretical probability distribution of a fair 12-sided die

$$ P(X = i) = \frac{1}{12} \approx 0.0833 \quad \text{for } i \in \{1, 2, \dots, 12\} $$

Calculate the expected frequency for each outcome in 180 rolls

$$ E(f_i) = 180 \times \frac{1}{12} = 15 \quad \text{for each outcome } i $$

Identify the histogram representing a uniform distribution with random variation

• Histogram A shows a strictly decreasing trend.
• Histogram B shows a symmetric, bell-shaped (normal) distribution.
• Histogram C shows frequencies fluctuating closely around the expected value of 15.
• Histogram D shows a strictly increasing trend.

Answer:

• A. The histogram showing a strictly decreasing trend from left to right.
• B. The histogram showing a symmetric, bell-shaped distribution centered around 6 and 7.
• C. The histogram showing fluctuating frequencies that are all relatively close to 15. (Correct answer)
• D. The histogram showing a strictly increasing trend from left to right.