Probability Distributions

Model the behavior of mathematical outcomes. Explore discrete distributions through rolling dice combinations, or Continuous Normal distributions shading areas under standard bell curves.

6x6 Dice Sum combinations space

1. Dice Rolling Simulator

Roll pairs of six-sided dice. Experimental counts populate as bars stacked over the theoretical triangular outline.

2. Selected Sum Stats

Choose Target Sum:

Theoretical Probability

P(7) = 6/36 = 16.67%

Experimental Probability

Exp. P = 0/0 = 0.00%

Total Rolls N = 0
μ - 3σ μ - 2σ μ - σ μ (Mean) μ + σ μ + 2σ μ + 3σ

1. Adjust Normal Density Bounds

Mean (μ): 5.0
Standard Deviation (σ): 1.5

2. Shaded Area Boundaries

Adjust boundaries a and b to calculate cumulative probability P(a ≤ X ≤ b):

Lower Bound (a): 3.5
Upper Bound (b): 6.5

Probability (Shaded Area)

P(3.5 ≤ X ≤ 6.5) = 68.27%

Algebraic Solver

Probability Density Walkthrough

Trace the mathematical resolution for the selected distribution and its bounds.

Trace Calculations Steps

Evaluation

Distributions & Densities Quiz

Test your conceptual knowledge of discrete coin/dice sums, continuous density shading, standard normal scaling, and area integrals.

Question 1 of 5 Score: 0/0

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Correct Answer!

Explanation text...

Distribution Key Rules


  • Discrete Probabilities: The outcomes are countable (e.g. sums of rolling dice). The sum of all probabilities in the sample space always equals exactly **1** (or 100%).
  • Continuous Probabilities: Outcomes represent ranges (e.g. heights). Probability is represented as shaded area under the density curve. The total volume/area under the curve is always **1.0**.
  • Z-Score Standardizing: Measures how many standard deviations a value $x$ lies from the mean $\mu$: $Z = (x - \mu)/\sigma$.
  • Empirical Rule (68-95-99.7): In any Normal distribution:
    • Approx. **68%** lies within $\pm 1\sigma$.
    • Approx. **95%** lies within $\pm 2\sigma$.
    • Approx. **99.7%** lies within $\pm 3\sigma$.