Probability quantifies uncertainty. P(Event) = Number of favorable outcomes / Total possible outcomes.
Key Concepts
- Independent events: P(A and B) = P(A) × P(B)
- Mutually exclusive: P(A or B) = P(A) + P(B)
- Conditional: P(A|B) = P(A and B) / P(B)
- Complement: P(not A) = 1 − P(A)
Permutations vs. Combinations
Permutations (order matters): nPr = n! / (n−r)!
Combinations (order doesn't matter): nCr = n! / [r!(n−r)!]
The Gambler's Fallacy
After a coin lands heads five times in a row, many people assume tails is "due" — but for a fair coin, each flip is independent, and the probability of heads on the next flip is still exactly 50%, regardless of past results. This misconception (the gambler's fallacy) causes real financial losses in gambling and poor decisions in investing.
Expected Value
Expected value multiplies each possible outcome by its probability and sums the results, giving the long-run average outcome. A lottery ticket with a 1-in-1,000,000 chance to win $500,000 has an expected value of $0.50 — meaning it's a bad bet at any price above 50 cents, even though any single ticket might "feel" worth the risk.