Probability Calculator
Calculate probability for basic events, combinations, permutations, dice, cards and coins
Example: rolling a number ≤ 3 on a 10-sided die → 3 favourable, 10 total
// Formula Reference
Understanding Probability
Probability measures how likely an event is to occur, expressed as a number between 0 (impossible) and 1 (certain). It can also be expressed as a percentage (0–100%) or as odds (e.g. 3:1 against).
Key Formulas
Combinations vs Permutations
Combinations count selections where order doesn't matter. Choosing 3 people from 10 for a committee — {A,B,C} is the same as {C,A,B}. Permutations count arrangements where order matters. Choosing a president, vice-president and secretary from 10 people — {A,B,C} and {C,A,B} are different outcomes.
Probability: The Five Rules That Solve Almost Everything
Built and verified by Andrius R. · Updated June 2026
Probability has a reputation for trickiness it only half deserves: most everyday problems fall to a handful of rules. The trickiness is real, though — human intuition is reliably wrong about randomness in specific, predictable ways. Both halves are below.
The core toolkit
- Basic definition: favorable ÷ total outcomes. One die showing 6: 1/6 ≈ 16.7%.
- AND (independent events) → multiply: two dice both showing 6: 1/6 × 1/6 = 1/36 ≈ 2.8%. Five coin heads in a row: (1/2)⁵ = 1/32 ≈ 3.1%.
- OR (mutually exclusive) → add: rolling a 1 or a 2: 1/6 + 1/6 = 1/3. (If events can overlap, subtract the overlap once.)
- NOT → one minus: the workhorse. P(at least one success) = 1 − P(zero successes).
- Dependence changes everything: drawing two aces without replacement is 4/52 × 3/51, not (4/52)² — the first draw changes the deck.
The "at least one" trick, demonstrated
Computing this directly means juggling overlapping cases. Flip it: P(no 6 in a roll) = 5/6, so P(no 6 in four rolls) = (5/6)⁴ ≈ 0.482. Therefore P(at least one 6) = 1 − 0.482 = ~51.8%.
Slightly better than a coin flip — which already breaks most people's intuition that four tries at a 1-in-6 event should feel closer to 4/6. Chances don't add across attempts; the complement multiplies.
The birthday paradox: intuition's most famous defeat
In a room of just 23 people, the probability that some two share a birthday is ~50.7%; at 50 people it's ~97%. The trap is that intuition counts comparisons to you (22 chances), while the math counts every pair — 23 people make 253 pairs, and 253 lottery tickets against odds of 1-in-365 each add up fast. It's the same complement trick: 1 − (365/365 × 364/365 × … × 343/365). The general lesson transfers everywhere: coincidences among many possible pairings are far more likely than they feel.
Where intuition predictably fails
- The gambler's fallacy: after five reds on a roulette wheel, black is not "due" — each spin is independent, and the wheel has no memory. (Its mirror twin, the hot-hand assumption, errs the same way in reverse.)
- Big numbers don't scale in the gut: a 1-in-14-million lottery jackpot vs 1-in-300 odds of some minor prize feel vaguely similar as words; they differ by a factor of ~47,000.
- Rare-event tests: a 99%-accurate test for a 1-in-1,000 condition still produces ~10 false positives for every true positive — base rates dominate accuracy, a result that surprises even clinicians.
- Long runs look non-random: genuine random sequences contain streaks (five heads in a row appears more often than people accept), which is why fabricated "random" data is often caught for being too evenly mixed.
Odds vs probability — and converting between them
Bookmakers speak odds; statisticians speak probability. "3-to-1 against" means 1 success per 4 total outcomes: probability 1/4 = 25%. Conversion: probability = odds-of-success ÷ (sum of both sides). Betting odds also carry the house's margin baked in — sum the implied probabilities of all outcomes of a real sportsbook event and you'll get over 100%; the excess is the bookmaker's edge, and it's why the expected value of betting is negative before any skill enters.
From the Blog
// 0 to 1 Scale
Probability is always between 0 (impossible) and 1 (certain). 0.5 = 50% chance. Anything above 1 or below 0 indicates an error.
// Complement Rule
It's often easier to calculate P(not happening) and subtract from 1. P(at least one 6 in 3 rolls) = 1 − P(no 6 in 3 rolls).
// Independent Events
For independent events, multiply probabilities. P(two heads) = P(head) × P(head) = 0.5 × 0.5 = 0.25.
// Gamblers Fallacy
Past outcomes don't affect future independent events. After 10 heads in a row, the next flip is still 50/50.