Probability from 0 to 1: addition and multiplication rules, dependent vs independent events, and expected value turning probabilities into long-run averages.
Your friend rolls two dice and offers you even money that at least one shows a six. Gut feeling says two dice out of six faces means a 1-in-3 shot, so even odds sound fair. The real probability is 11/36 — about 30.6% — and over a long enough session you lose money on that bet. Probability is the branch of mathematics that settles exactly this kind of disagreement between intuition and reality, and its core rules are simple enough to fit on an index card.
Probability is a way to measure uncertainty on a scale from 0 to 1. A probability of 0 means an event never happens, a probability of 1 means it always happens, and values in between describe how likely it is. We often convert these decimals into percentages by multiplying by 100.
The most direct way to calculate probability is to count the outcomes you want and divide by all possible outcomes. This assumes every outcome is equally likely, which is true for fair coins, dice, and cards but not for everything in real life.
P(event) = Number of Favorable Outcomes / Total Number of Possible Outcomes
For a fair six-sided die, the probability of rolling a 4 is 1 favorable outcome out of 6 possible outcomes, or 1/6, about 0.167. The probability of rolling an even number is 3 favorable outcomes (2, 4, 6) out of 6, which simplifies to 1/2, or 0.5.
Three rules handle almost every probability calculation. The complement rule says the probability of an event not happening is 1 minus the probability that it does. The addition rule combines probabilities of either of two events happening, and the multiplication rule combines probabilities of both happening.
P(not A) = 1 - P(A)
P(A or B) = P(A) + P(B) - P(A and B)
P(A and B) = P(A) x P(B) (independent events)
The addition rule subtracts the overlap so events that satisfy both conditions are not counted twice. If there is a 30% chance of rain and a 20% chance of wind, with a 10% chance of both, the chance of at least one is 0.30 plus 0.20 minus 0.10, equal to 0.40 or 40%.
The multiplication rule as written applies to independent events, where one outcome does not change the odds of another. For dependent events you instead multiply by the conditional probability of the second event given the first, which is the foundation of more advanced topics like Bayes theorem.
Expected value is the long-run average outcome of a random process. It does not predict any single result, but it tells you what to expect on average if you repeated the process many times. To compute it, multiply each possible outcome by its probability and sum the results.
E(X) = Sum of [ Outcome x P(Outcome) ]
Consider a raffle ticket that wins a $100 prize with a 1 in 200 chance. The expected value of one ticket is $100 multiplied by 1/200, which is $0.50. If the ticket costs more than $0.50, buying it has a negative expected value on average, even though someone will win.
Expected value is how casinos and insurers stay profitable. Every individual bet or policy is uncertain, but across thousands of customers the average always drifts toward the expected value. Understanding it helps you evaluate gambles, investments, and insurance deductibles on their true long-run merits.
Probability shows up everywhere once you look for it. Flipping a fair coin has a 0.5 chance of heads, and because flips are independent, the chance of two heads in a row is 0.5 times 0.5, or 0.25. The chance of rolling at least one 6 in two dice is 1 minus the chance of no sixes, which is 1 minus (5/6 times 5/6), about 0.306.
Weather forecasts are probabilities too. An 80% chance of rain does not mean it will rain for 80% of the day, it means that under similar historical conditions it rained about 80% of the time. Reading forecasts this way makes them far more useful for planning.
Lotteries illustrate extreme probability. The odds of winning a typical 6-from-49 jackpot are about 1 in 13.98 million, a probability near 0.00000007. Combined with the expected value framework, this explains why lotteries are entertainment rather than investment: the expected return is almost always far below the ticket price.
Conditional probability asks how the chance of one event changes when you know another event happened. The formula is P(A given B) = P(A and B) / P(B). The division is what makes conditional probability counterintuitive: the answer depends on how common the conditioning event is in the first place, not just on how accurate the test or signal appears to be.
Medical screening is the clearest example. Suppose a disease affects 1 in 1,000 people, and a screening test catches 99% of true cases (sensitivity) while keeping false positives at 5% of healthy people (so its specificity is 95%). Out of 100,000 people: 100 actually have the disease, and the test flags 99 of them. But the test also flags 5% of the 99,900 healthy people — about 4,995 false positives. So among the 5,094 positive results, only 99 are real. A positive result means a 99/5,094 ≈ 1.9% chance of actually having the disease, even though the test is "99% accurate."
This is called the base rate fallacy, and it is not a hypothetical — it shapes how people interpret screening results, fraud alerts, and spam filters. When an event is rare, even a highly accurate test produces mostly false positives, because the healthy population dwarfs the affected one. Whenever you evaluate a percentage attached to a signal, ask what fraction of the underlying population actually has the property you are testing for.
The Probability Calculator on this site implements all of these cases — single events, AND and OR combinations, complements, and expected value — and shows the formula it used next to the result, so you can check the reasoning rather than trust the answer.
Probability Calculator — Odds, Success & EV
Put the numbers from this guide to work. The calculation runs entirely in your browser — nothing is sent to a server.