Probability Calculator

The Basic Probability Formula

Probability measures how likely an event is, on a scale from 0 (impossible) to 1 (certain) — often written as a percentage. For a single event with equally likely outcomes, it's a simple ratio:

Rolling a 4 on a standard six-sided die: there's 1 favorable outcome (the face showing 4) out of 6 total possible outcomes, so P(4) = 1/6 ≈ 16.67%. This only works cleanly when every outcome is equally likely — a loaded die or a biased coin needs a different approach (the actual observed frequency, not a theoretical count).

Combining Two Events: AND vs. OR

Real problems usually involve more than one event, and the two most common questions are "what's the chance BOTH happen" and "what's the chance AT LEAST ONE happens":

• P(A and B), independent events: P(A) × P(B) — e.g. flipping heads twice in a row: 0.5 × 0.5 = 0.25.
• P(A or B), independent events: P(A) + P(B) − P(A and B) — you subtract the overlap so it isn't double-counted.
• P(A or B), mutually exclusive events: just P(A) + P(B) — since the events can't both happen, there's no overlap to subtract (e.g. rolling a 2 or a 5 on one die roll — you can't roll both at once).

Mixing these up is the most common mistake — using the mutually-exclusive OR formula on events that can actually overlap will overstate the probability, since you haven't removed the double-counted overlap.

How to Use This Calculator

• Single Event tab: enter favorable outcomes and total possible outcomes for a straightforward ratio.
• Two Events tab: enter P(A) and P(B) as decimals (0 to 1) — tick "mutually exclusive" if the two events can't both happen.
• Complement tab: enter P(A) to instantly get P(not A) = 1 − P(A).
• For repeated trials with a fixed success probability (coin flips, pass/fail attempts), use the dedicated Binomial Probability Calculator instead — it handles multi-trial scenarios this tool doesn't.

Theoretical vs. Experimental Probability

Everything above is theoretical probability — calculated in advance from counting equally-likely outcomes, assuming a fair die, an unbiased coin, or a well-shuffled deck. Experimental (or empirical) probability instead comes from actually running a trial many times and counting what happened: flip a coin 100 times, count 47 heads, and the experimental probability of heads in that run was 0.47, not the theoretical 0.5. With enough repeated trials, experimental probability tends to converge toward the theoretical value — a pattern known as the Law of Large Numbers — but any single small run of trials can deviate meaningfully from theory purely by chance, which is why "the coin lands heads 3 times in a row" doesn't mean the coin is unfair, and why real-world estimates (like a marketing conversion rate, or a manufacturing defect rate) are usually built from experimental data rather than a theoretical assumption.

A Classic Trap: The Gambler's Fallacy

One of the most common probability mistakes is assuming that independent events somehow "even out" in the short run — believing a coin is "due" for tails after several heads in a row, or that a roulette wheel is more likely to land on black after a run of reds. For genuinely independent events, each trial's probability is completely unaffected by what happened before it: a fair coin is exactly 50% likely to land heads on the next flip regardless of the last ten flips' results. This mistake, known as the gambler's fallacy, is worth watching for specifically because it feels intuitively correct — the math above (P(A and B) = P(A) × P(B) for independent events) is precisely what shows why each individual flip's probability never changes based on prior outcomes.

Frequently Asked Questions