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Weighted Expected Value Visualization



Higher probabilities "pull" the expected value toward them. Watch how E(X) differs from the simple average when probabilities are unequal.

1P=0.171×0.17=0.172P=0.172×0.17=0.333P=0.173×0.17=0.504P=0.174×0.17=0.675P=0.175×0.17=0.836P=0.176×0.17=1.00E(X) = 3.50Avg = 3.50Probability "Weights" Pull the Expected ValueBlue numbers show each contribution: x × P(x) = contribution to E(X)P(X = x) in circlePull strength (arrow)Expected Value E(X)Simple Average (unweighted)

Calculation

Expected Value (Weighted):
1 × 0.17 = 0.17
2 × 0.17 = 0.33
3 × 0.17 = 0.50
4 × 0.17 = 0.67
5 × 0.17 = 0.83
6 × 0.17 = 1.00
E(X) = 3.500
Simple Average (Unweighted):
(1 + 2 + 3 + 4 + 5 + 6) / 6
Avg = 3.500

Notice: When probabilities are equal, E(X) = Avg. When probabilities differ, E(X) is pulled toward the high-probability values.

Understanding Weighted Average

  • Blue circles contain P(X = x) - the probability of each outcome. Circle size also shows probability
  • Arrow thickness/length shows "pull strength" - how much that outcome pulls E(X) toward it
  • Red line (E(X)) is the weighted average - pulled toward high-probability outcomes
  • Gray dashed line is the simple average (unweighted) - treats all outcomes equally
Key Insight:

When probabilities are equal, E(X) = simple average. When probabilities differ, E(X) is pulled toward high-probability outcomes. This is why it is called a weighted average!


Weighted Averages
Visual Representation
Interactive Exploration



Understanding Weighted Averages in Probability

Expected value is a weighted average where each outcome is weighted by its probability of occurrence. This visualization shows how different probability weights affect the final expected value.



Weighted Averages

Unlike a simple average where all values have equal weight, expected value gives more weight to outcomes with higher probabilities. This reflects their greater influence on the long-run average.
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Visual Representation

The weighted visualization shows each outcome as a bar whose contribution to the expected value is proportional to both the outcome value and its probability.
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Interactive Exploration

Adjust probabilities and outcomes to see how changes affect the expected value. Notice how high-probability outcomes have greater influence on E[X].
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