How probability works
Probability is a number between 0 and 1 that measures how likely something is to happen. A probability of 0 means it is impossible; a probability of 1 means it is certain. Anything in between is uncertain to a degree we can quantify. The classical formula is simple: count the number of favorable outcomes, count the total number of possible outcomes, and divide. If you want the chance of drawing a heart from a standard deck, there are 13 hearts out of 52 cards, so the probability is 13 ÷ 52 = 0.25, or 25%.
Single vs multiple events
For a single event, the formula above is enough. For multiple events, the rules depend on whether the events are independent (one outcome doesn't affect another) and whether you want them to occur together (AND) or at least one of them (OR).
- AND (intersection): P(A and B) = P(A) × P(B) for independent events. Roll a die twice and want two sixes? 1/6 × 1/6 = 1/36.
- OR (union): P(A or B) = P(A) + P(B) − P(A and B). Subtracting the overlap prevents double counting.
- Complement: P(not A) = 1 − P(A). Useful when calculating "at least one" probabilities — find the chance it never happens, then subtract from 1.
Conditional probability
Conditional probability asks: given that B has already happened, what is the chance of A? The formula is P(A|B) = P(A and B) ÷ P(B). This is the foundation of Bayesian reasoning, medical test interpretation, and machine learning. A test that is "95% accurate" can still produce mostly false positives if the disease is rare — Bayes' theorem reveals why.
Dice and coin probability
Rolling two six-sided dice gives 36 equally likely outcomes (6 × 6). Of those, exactly six combinations sum to 7, so the probability of rolling a 7 is 6 ÷ 36 ≈ 16.67%. The probability of rolling 2 or 12 is just 1 ÷ 36 each, because only one combination produces each. The shape of dice-sum probability is a triangle that peaks at the average roll.
Coin flips follow the binomial distribution. The chance of getting exactly k heads in n flips of a fair coin is C(n,k) × 0.5n. With 10 flips, the most likely number of heads is 5 (probability about 24.6%), but you only get exactly that result less than a quarter of the time — variability is the rule, not the exception.
Combinations and permutations
Combinations count selections where order doesn't matter — a five-card poker hand is the same hand regardless of the order you receive the cards. Permutations count arrangements where order matters — first, second, and third place in a race are distinct outcomes. Combinations are always smaller than (or equal to) the corresponding permutations because the same set can be arranged in multiple orders.
Common probability mistakes
Even people who use probability daily fall for a few classic traps. The gambler's fallacy assumes that past outcomes affect independent future ones — but a coin has no memory, and a streak of heads doesn't make tails "due." The base rate fallacy ignores how rare an event is in the general population, leading to overconfidence in noisy tests. And conjunction fallacy mistakenly judges a specific scenario as more likely than a general one because it sounds more believable.
Use this calculator to build intuition: change the inputs, watch the math, and notice how small probabilities multiply quickly into very small ones, while OR probabilities saturate as you add more chances. Probability rewards careful counting, not gut feelings.
Try also
- Dice Roller — simulate rolls to see probability in action
- Coin Flipper — flip coins one at a time or in batches
- Random Number Generator — pick numbers in any range
- Scientific Notation Converter — express tiny probabilities clearly