Odds and Probability
Counting the ways an outcome can happen
Three fair coin tosses make eight patterns. Exactly two heads is three of them, or 3/8, not a specific order and not a system.
Odds and Probability
Odds and Probability
Probability trees for a two-step game
A probability tree multiplies along a path and adds across paths. Two hearts from a deck show why the second chance shrinks.
Odds and Probability
Mutually exclusive events and the addition rule
Mutually exclusive events can be added. Overlapping events cannot, unless you subtract the double count. A die shows both cases.
Odds and Probability
Coin-flip streaks and what they do not prove
Five heads in a specified row is 1 in 32 on a fair coin. The run does not schedule the next toss or prove a system.
Odds and Probability
Standard deviation in everyday language
Standard deviation measures typical scatter around an average. A fair-coin example shows why 100 small bets still swing.
Odds and Probability
House edge as a long-run price, not a forecast
House edge is the long-run loss as a share of money wagered. A checked roulette example shows the price, not a forecast.
Odds and Probability
Conditional probability with a two-card draw
A condition changes the denominator. Two cards from a 52-card deck show how an ace already dealt changes the next chance.
Odds and Probability
Why a parlay is less likely than it feels
A parlay pays only if every leg hits. Independent chances multiply downward, and the posted price rarely pays that product back.
Odds and Probability
Combinations and permutations with cards
Permutations count order. Combinations do not. A card-counting lesson in the arithmetic sense, with the poker-hand figure checked.
Odds and Probability
Variance: why results swing before they settle
Variance is the spread around an average. It explains wild sessions without turning a streak into a prediction.
Odds and Probability
The law of large numbers without jargon
The law of large numbers says averages settle. It does not say the next result will cancel the last one.
Odds and Probability
What independent events actually mean
Independent events do not remember the past. See what that means for coins, dice and cards, and what it does not mean.