Probability Calculator
Enter your event probabilities and get an instant result — as a decimal, percentage, and fraction — with a plain-language explanation of what the number actually means.
Two Event Probability
Binomial Probability (Successes in Trials)
How to read your result
Probability is expressed several ways. They all mean the same thing.
Decimal (0 to 1)
The standard mathematical form. 0 means impossible, 1 means certain. A result of 0.25 means the event happens one time in four, on average.
Percentage
The decimal multiplied by 100. Most people find percentages more intuitive — 25% is easier to reason about than 0.25. Both are equally correct.
Fraction
Shows the ratio directly. 1/4 means one favourable outcome out of four possible outcomes. Fractions are especially useful in card and dice problems.
Odds (X to Y)
Odds express the ratio of favourable to unfavourable outcomes. A 25% probability is 1:3 odds — one win for every three losses. Common in gambling and betting contexts.
Independent vs. dependent events
The most important question to ask before any probability calculation: does the first event change the conditions for the second?
Independent events
Also called: events with replacement
Each trial is completely fresh. A coin doesn't remember its last flip. Drawing a card, writing down its suit, putting it back, and shuffling before drawing again — that's independent. The pool never changes.
Example: two heads in a row = 0.5 × 0.5 = 0.25
Dependent events
Also called: events without replacement
The outcome of the first trial changes the odds for the next. Draw an ace from a 52-card deck and keep it — there are now only 3 aces left in a 51-card deck. The denominator has shifted.
P(B | A) means "probability of B given A has already occurred"
Mutually exclusive events
Two events are mutually exclusive when both can't happen at the same time. Rolling a 2 and rolling a 5 on a single die throw — you get one or the other, never both. Being in London and in Tokyo simultaneously — mutually exclusive.
For mutually exclusive events, the probability of either one happening is simply the sum of their individual probabilities. Roll a 2 or a 5 on a fair die: 1/6 + 1/6 = 2/6 ≈ 33.3%.
Compare that to non-exclusive events, where both can happen at once. If you're asking "what's the probability of drawing a red card or a king?" — some kings are red, so you'd double-count without the inclusion-exclusion correction: P(A) + P(B) − P(A and B).
Where this calculator is actually useful
Probability comes up in more places than you'd expect — not just statistics class.
Games and gambling
What are the odds of rolling two sixes back to back? What's the chance of being dealt a pair in poker? Any game involving dice, cards, or random draws is a probability problem. Knowing the exact odds helps you make rational decisions instead of relying on instinct.
Science and research
Researchers use probability to determine whether an experiment's results are meaningful or just random noise. A drug trial, a physics experiment, a quality-control test — all involve asking "how likely is this result if nothing real is happening?"
Business and risk
What's the probability that at least one of three suppliers fails this quarter? What are the odds two independent systems both go down at the same time? Operations, finance, and project management all reduce to probability questions like these.
Homework and exam prep
Probability is one of the more confusing areas of secondary and university maths because intuition fails so often. This calculator lets you check your working, explore what happens when you change the numbers, and build a feel for how the formulas behave.
Everyday reasoning
A 30% chance of rain sounds low — but if you need it dry three days in a row, the combined probability of all three being dry is 0.7³ = 34%. Understanding compound probabilities changes how you plan.
Worked examples
Three common scenarios with the full working shown.
Two independent events — coin and die
What's the probability of flipping heads AND rolling a 6?
P(heads) = 1/2 = 0.5P(rolling 6) = 1/6 ≈ 0.167P(both) = 0.5 × 0.167 = 0.0833Result: 8.3% — happens roughly once in every 12 attempts.
Mutually exclusive — rolling a 1 or a 6
What's the probability of rolling a 1 or a 6 on one throw?
P(rolling 1) = 1/6P(rolling 6) = 1/6P(1 or 6) = 1/6 + 1/6 = 2/6Result: 33.3% — one in three throws will land on 1 or 6.
Dependent events — drawing cards without replacement
What's the probability of drawing two aces from a 52-card deck without replacing the first?
P(first ace) = 4/52 ≈ 0.077P(second ace | first ace removed) = 3/51 ≈ 0.059P(both) = 0.077 × 0.059 = 0.0045Result: 0.45% — happens roughly once in every 221 two-card draws.
The Gambler's Fallacy
In 1913, the roulette wheel at the Monte Carlo Casino landed on black 26 consecutive times. As the streak grew, players bet increasingly large sums on red — convinced that red was "due." The streak eventually ended, but not before enormous losses. The mathematical reality: on spin 27, the probability of black was still 47.4%, completely unaffected by the previous 26 spins.
This is the Gambler's Fallacy — the intuition that random independent events somehow "balance out" over time. They don't. A roulette wheel has no memory. A coin has no memory. Each trial starts fresh. The past tells you nothing about the next outcome when events are independent.
The fallacy is so persistent because our brains are pattern-matching machines. Seeing ten heads in a row feels like it "means something." It doesn't — it's just a sequence with a 1 in 1,024 chance that had to happen to someone, and today it happened to you.
Common mistakes to avoid
- —Adding probabilities when you should multiply. "Both events happen" means multiply. "Either event happens" (and they're mutually exclusive) means add.
- —Forgetting to subtract the overlap. If events aren't mutually exclusive, P(A or B) = P(A) + P(B) − P(A and B). Skip the subtraction and you'll overcount.
- —Treating dependent events as independent. Drawing cards without replacement changes the odds each time. Always check whether the sample space is shrinking.
- —Confusing probability with odds. A 1/4 probability (25%) is 1:3 odds — one win, three losses. They're related but not the same number.
- —Assuming rare events won't happen. A 1% chance per trial means you'll see it happen, on average, within 100 trials — and sometimes much sooner.
Frequently asked questions
Why is probability always between 0 and 1?
Because it represents a proportion of all possible outcomes. If something is impossible, zero out of every possible outcome is favourable — that's 0. If something is certain, every possible outcome is favourable — that's 1. You can't have more than all outcomes, so you can't go above 1.
What's the difference between mutually exclusive and independent?
They describe different things. Mutually exclusive means the events can't both happen at the same time (rolling a 2 and rolling a 5 in one throw). Independent means one event doesn't affect the probability of the other (coin flip doesn't affect die roll). Events can be one, both, or neither.
How do I calculate the probability that at least one of several events happens?
The complement approach is easiest: P(at least one) = 1 − P(none happen). Find the probability that every event fails, then subtract from 1. For example, if three independent events each have a 10% chance, P(at least one) = 1 − (0.9 × 0.9 × 0.9) = 1 − 0.729 = 27.1%.
Does this calculator handle binomial probability?
Binomial probability covers situations where you run the same trial multiple times and want to know how likely a specific number of successes is. For example: if I flip a coin 10 times, what's the probability of exactly 6 heads? Check the calculator's mode options — binomial distribution may be available as a separate input mode.
Is my data stored when I use this?
No. All calculations run locally in your browser. Nothing is transmitted to a server or stored in any way. Your inputs disappear when you close the tab.