Probability Calculator
Pick the scenario below that matches your problem, a coin flip, a card draw, a positive test result, a repeated trial, and get the answer worked out step by step, with every number shown.
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The rest of the toolkit
Each calculator below lives on its own page, with more depth and worked examples than fits in the tool above.
Conditional Probability Calculator
Finds P(A|B) straight from a two-way table of counts as well as from raw probabilities, so you're not stuck converting a textbook table into decimals by hand first.
Bayes' Theorem Calculator
Breaks a positive test result down across a full sample population, true positives, false positives, and all, so the answer shows its work instead of asking you to trust one formula.
Binomial Probability Calculator
Covers exactly, at least, and at most in one pass, and charts the full distribution so you can see how your target outcome compares to everything else that could have happened.
Combinations and Permutations Calculator
Asks two plain questions, does order matter, can items repeat, and picks the right one of four formulas for you, instead of expecting you to already know which term applies.
Dice Probability Calculator
Splits single-die odds, two-dice sums, and multi-roll streaks into separate modes, since those are three genuinely different questions that get lumped together everywhere else.
Coin Flip Probability Calculator
Tells one exact sequence of flips apart from any arrangement with the same head count, the single mix-up behind most wrong answers to coin flip questions.
Card Probability Calculator
Runs on the hypergeometric formula that dealt hands actually need, rather than the binomial shortcut that looks close enough but quietly gets the odds wrong.
Birthday Paradox Calculator
Finds the exact chance that two people in a group share a birthday, plus how many people you'd need for any target odds, and why 23 is already enough for better than even.
Lottery Odds Calculator
Works out the real odds for Powerball, EuroMillions, UK Lotto or any custom format, including matching fewer numbers and what buying extra tickets actually does to your chances.
Monty Hall Problem Calculator
Play the classic three-door puzzle yourself, then see the exact math behind why switching wins twice as often as staying, with a version for any number of doors.
Sensitivity and Specificity Calculator
Turns a test's sensitivity and specificity into positive and negative predictive value, likelihood ratios, and a full ROC curve, straight from rates or a results table.
The kinds of probability problem, and how to tell them apart
Most probability confusion doesn't come from the math. It comes from not knowing which formula applies. Here's the practical difference between the cases this page handles.
A single event asks about one outcome out of a fixed set of possibilities: what's the chance of pulling a red card, rolling a 4, or picking the winning raffle ticket. You count the outcomes you want and divide by all the outcomes that could happen.
Independent events come up when two things happen and neither one changes the odds of the other. Flipping a coin twice is the textbook case. You multiply the two probabilities straight through.
Dependent events are trickier because the first outcome changes what's possible for the second. Draw a card and don't put it back, and there are only 51 cards left for the next draw.
Conditional probability is the plain question "given B happened, what's the chance A also happened?" It's useful when you already know the joint probability P(A∩B) rather than P(B|A).
Bayes' Theorem updates a belief once new evidence shows up: start with a baseline probability, get a piece of information, and recalculate.
Repeated trials (binomial) answer questions like "3 heads in 5 flips" or "at least one success in 10 tries." It's the same trial repeated a fixed number of times with the same success probability each time.
Combinations and permutations count arrangements rather than probabilities directly, but they're the building blocks behind lottery odds, hand-of-cards odds, and any "how many ways can this happen" question.
Odds express the same information as probability in a different, betting-style format, like "3 to 1 against" instead of "25% chance."
The formulas, with an explanation of where each one comes from
A worked example that shows why test accuracy alone doesn't tell you much
Say a disease affects 1 in 100 people, and a test correctly flags 95% of people who have the disease and wrongly flags 5% of people who don't. Someone tests positive. What's the actual chance they have the disease?
Most people guess somewhere around 90 to 95%. The real answer is closer to 16%.
Imagine 10,000 people take the test. 1% actually have the disease, which is 100 people, and the test correctly flags 95 of them. Among the 9,900 healthy people, 5% still test positive by mistake: 495 false alarms. That's 590 positives in total, but only 95 are actually sick. 95 divided by 590 is roughly 16%.
This is exactly what Bayes' Theorem calculates, and it's why doctors run a second, different test before treating a positive result as conclusive when the underlying condition is rare.
Mistakes people make with probability that are easy to avoid
Treating dependent events as independent. Drawing two cards without replacement is dependent: the second draw has one fewer card, and one fewer of whatever came out first.
Forgetting the subtraction in the addition rule. If P(A) = 0.6 and P(B) = 0.7, simply adding them gives 1.3, an impossible probability.
Reading "95% accurate" as "95% chance you have it." As the worked example above shows, those are different numbers once you account for how rare the condition actually is.
Confusing P(A|B) with P(B|A). The probability of rain given clouds is not the same as the probability of clouds given rain. Mixing these up is the base rate fallacy.
Assuming "at least one" scales linearly. With a 20% chance per attempt, the chance of at least one success in 5 attempts isn't 100%. It's 1 − (0.8)5 ≈ 67%, which the Repeated Trials calculator above works out for you.
Questions people actually ask about this
What's the difference between P(A∩B) and P(A∪B)?
P(A∩B) is both things happening together. P(A∪B) is either one happening, or both. Mixing these up is the single most common notation error in intro probability.
Why can't probability be negative or above 1?
Probability is a proportion of outcomes out of a total. If your calculation lands outside 0 to 1, check your inputs. Most often it means you added two probabilities without subtracting their overlap.
How do I check whether two events are independent?
Test whether P(A∩B) equals P(A) × P(B). In practice, independence is usually assumed from how the scenario is described (separate coin flips, drawing "with replacement") rather than checked directly.
What's the difference between a combination and a permutation?
A combination doesn't care about order, like picking 3 lottery numbers from a pool. A permutation does care about order, like awarding gold, silver and bronze from the same pool of runners. nPr is always at least as large as nCr for the same n and r.
How do I convert "3 to 1 odds" into a probability?
Odds of A:B in favour convert to probability A / (A+B). So 3 to 1 in favour is 3/(3+1) = 0.75. Odds "against" simply flip which side is which.
Is a 1-in-a-million chance the same as 0.0001%?
No. 1 in a million is 0.0001 as a decimal, which is 0.0001 × 100 = 0.01%, not 0.0001%.
Why did the dependent-events calculator reject my numbers?
It checks that P(A), P(B), and P(B|A) are logically consistent before computing anything. For example, P(A∩B) can never be larger than either P(A) or P(B) on its own, since the intersection of two events is always a subset of each one.