How Probability Works
What Is Probability?
Flip coins and watch randomness converge to a pattern
With very few flips the ratio bounces wildly — randomness dominates at small sample sizes.
Step 1 of 6: What Is Probability?
Flip coins and watch randomness converge to a pattern
With very few flips the ratio bounces wildly — randomness dominates at small sample sizes.
Step 2 of 6: The Bell Curve
Reshape a normal distribution with mean and standard deviation
The bell curve is symmetric around μ. About 68% of values fall within ±1σ of the mean.
Step 3 of 6: Uniform & Bernoulli
Two fundamental building-block distributions
Near p = 0.5, both outcomes are equally likely and variance is maximized.
Step 4 of 6: Joint & Conditional
When two events interact — Bayes' theorem in a Venn diagram
P(A∩B) = P(A)·P(B|A) is the multiplication rule. Bayes' theorem inverts this to get P(A|B) = P(A∩B)/P(B).
Step 5 of 6: Expected Value
The probability-weighted average that summarizes a distribution
E[X] is the probability-weighted average. It's the single number that best summarizes where outcomes tend to land.
Step 6 of 6: Why It Matters for ML
More data means better estimates — the foundation of learning
As the dataset grows, the sample mean converges toward the true mean. In ML, more training data yields more reliable gradient estimates.