The bell curve describes how data points cluster around a mean value, with predictable proportions falling within standard deviations.
f(x) = (1/σ√2π) e-(x-μ)²/2σ²
Update beliefs when new evidence arrives. Prior probability combined with likelihood gives you a more accurate posterior probability.
P(A|B) = P(B|A) · P(A) / P(B)
Use random sampling to estimate mathematical quantities. Generate thousands of random points to approximate areas, integrals, and probabilities.
π ≈ 4 · (points inside) / (total points)
As the number of flips increases, the ratio converges toward the true probability of 0.5 — the law of large numbers in action.
Roll distribution visualized as a histogram. Each face should appear approximately 16.7% of the time with a fair die.
Overlay two normal distributions to visually compare how changes in mean and variance affect shape and overlap. Observe how distributions diverge or converge.
A test is 99% accurate. You test positive for a rare disease (1 in 10,000). What is the actual probability you have the disease? The answer is surprisingly low — about 1% — because of base rate neglect.
How many people do you need in a room for a 50% chance that two share a birthday? Only 23. Our intuition about probability often fails us at combinatorial scale.
Behind three doors: one car, two goats. You pick a door, the host opens another revealing a goat. Should you switch? Yes — switching gives you a 2/3 probability of winning.
Pick a door to start.
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
P(A ∩ B) = P(A) · P(B|A)
P(A') = 1 − P(A)
P(A|B) = P(A ∩ B) / P(B)
E[X] = Σ xᵢ · P(xᵢ)
Var(X) = E[X²] − (E[X])²