Expectation

Watch how the running mean converges to the expected value E[X] as you roll the die. Adjust the probability distribution to create a biased die and observe how it affects the expectation.

Speed (ms):

Expected Value Visualization

Running mean convergence (left) and probability distribution (right)

Current Roll

Dice Distribution

Adjust Face Weights

Change weights to create a biased (unfair) die. Higher weight = more likely to roll.

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Expected Value

Expected Value E[X]

The probability-weighted average of all possible outcomes:

E[X] = Σ x·P(x)

For a fair die: E[X] = (1+2+3+4+5+6)/6 = 3.5

Law of Large Numbers

As you roll more times, the running mean converges to E[X]. This fundamental theorem connects theoretical probability to practical observation.

Variance

Probability

Measure statistical spread and distribution

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Chance Events

Probability

Explore random events and probability

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Standard Scaler

Preprocessing

Feature standardization (z-score)

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