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.

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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