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Probability Calculator
A probability calculator works out the odds for two independent events: the chance both happen, the chance at least one happens, and the chance neither happens. Enter each event's probability as a decimal from 0 to 1, and it shows every result as a decimal and a percent — useful for games, homework, and everyday decisions about risk. Below it, quick presets give the exact chances for a coin flip, dice rolls, and a card draw, and a second calculator finds the chance of at least one success in n trials.
A decimal from 0 to 1. For example, 0.5 means a 50% chance.
A decimal from 0 to 1. For example, 0.25 means a 25% chance.
| P(A and B) — both happen | |
| P(A or B) — at least one happens | |
| P(neither) — neither happens | |
| P(A only) | |
| P(B only) |
Quick presets
Tap a preset to see its exact chances — no simulation, just the math.
At least one success in n trials
Same chance every trial, independent trials — for example, flipping a coin 3 times, or rolling a die 5 times.
| P(at least one success) | |
| P(no successes at all) | |
| P(success every time) |
How it works
Probability measures how likely something is, on a scale from 0 to 1. A probability of 0 means the event cannot happen, 1 means it is certain, and 0.5 means it is a coin flip — equally likely to happen or not. Multiply by 100 and you get the same chance as a percent: 0.25 is 25%, 0.75 is 75%.
This calculator handles two independent events, which means one event does not change the chance of the other. Flipping two coins is independent, and so is rolling two dice, because the first result has no effect on the second. Drawing two cards from a deck without replacing the first is not independent, because the first card changes what is left — this calculator is not for that case, and using it there would give the wrong answer.
P(A or B) = P(A) + P(B) − P(A and B)
P(neither) = (1 − P(A)) × (1 − P(B)) = 1 − P(A or B)
P(A only) = P(A) × (1 − P(B)) • P(B only) = P(B) × (1 − P(A))
Why multiply for "and"? Because both things must happen, and each one narrows the chances. If each coin has a 50% chance of heads, the chance of two heads in a row is 0.5 × 0.5 = 0.25, or 25% — one outcome in four. The "or" formula adds the two chances, then subtracts the overlap where both happen, because that overlap was counted twice: with two coins, at-least-one-heads is 0.5 + 0.5 − 0.25 = 0.75, or 75%. "Neither" is the mirror image: the chance the first event fails times the chance the second fails.
All five results always add up in a tidy way: P(both) + P(A only) + P(B only) + P(neither) = 1, because those four outcomes cover everything that can happen. Checking that the four outcomes sum to 1 is a good way to confirm your answers, and the calculator's table makes it easy to see. If your numbers do not add to 1, something went wrong — though with this calculator, they always will.
The same multiplication idea answers a question that comes up constantly: what is the chance of at least one success if you try n times? The trick is to flip the question around — first find the chance of failing every single time, (1 − p)^n, then subtract from 1. A coin flipped 3 times has a 1 − 0.5³ = 1 − 0.125 = 0.875, or 87.5%, chance of at least one heads. The presets cover the classic cases directly: a coin flip (two heads in a row is 25%), one die (rolling a 6 is 1 in 6, about 16.67%), two dice (a total of 7 is the most likely roll at 6 in 36, also 16.67%), and a card from a standard 52-card deck (an ace is 4 in 52, about 7.69%, a heart is 13 in 52, or 25%).
Probability quietly shapes everyday choices. A weather forecast of a 30% chance of rain on Saturday and a 40% chance on Sunday means the chance of rain on both days is 0.3 × 0.4 = 12%, while the chance of at least one rainy day is 0.3 + 0.4 − 0.12 = 58%. A factory that knows each machine fails 5% of the time can work out the chance that at least one of two machines fails, and plan spare parts around it. The math is the same every time: multiply for "and", add-and-subtract-the-overlap for "or".
Worked example
Alex in Portland, Oregon, flips two fair coins and wants to know the chances for the different outcomes. Each coin has a 0.5 probability of landing on heads, so P(A) = 0.5 and P(B) = 0.5.
Step 1 — both heads: P(A and B) = 0.5 × 0.5 = 0.25 (25%). Step 2 — at least one heads: P(A or B) = 0.5 + 0.5 − 0.25 = 0.75 (75%). Step 3 — neither heads: P(neither) = (1 − 0.5) × (1 − 0.5) = 0.5 × 0.5 = 0.25 (25%). As a check, the four outcomes — both heads (0.25), first only (0.25), second only (0.25), neither (0.25) — add up to 1.00, as they must.
Enter P(A) = 0.5 and P(B) = 0.5 in the calculator above and you get the same result: P(both) = 0.25, P(at least one) = 0.75, P(neither) = 0.25.
FAQ
What is the chance of at least one success in n tries?
Work out the chance of failing every time and subtract it from 1: P(at least one) = 1 - (1 - p)^n, where p is the chance of success on one try and n is the number of tries. For a coin with p = 0.5 flipped 3 times, that is 1 - 0.5^3 = 1 - 0.125 = 0.875, or 87.5%.
What is the probability of drawing an ace from a deck of cards?
There are 4 aces in a standard 52-card deck, so the chance is 4/52, about 0.0769 or 7.69%. Drawing a heart is 13/52, or 25%, and drawing a face card (jack, queen, or king) is 12/52, about 23.08%.
What does independent mean in probability?
Two events are independent when one does not change the chance of the other. Flipping two coins is independent: the first coin landing on heads does not affect the second. Drawing cards without putting the first one back is not independent, because the first draw changes what is left in the deck.
How do you find P(A and B) for independent events?
Multiply the two probabilities: P(A and B) = P(A) x P(B). For example, if P(A) = 0.5 and P(B) = 0.5, then P(both) = 0.5 x 0.5 = 0.25, or 25%. This only works when the events are independent.
How do you find P(A or B)?
Add the probabilities and subtract the chance of both happening, so you do not count the overlap twice: P(A or B) = P(A) + P(B) - P(A and B). With P(A) = 0.5 and P(B) = 0.5, that is 0.5 + 0.5 - 0.25 = 0.75, or 75%.
What is the probability that neither event happens?
Multiply the chances that each event does not happen: P(neither) = (1 - P(A)) x (1 - P(B)). With P(A) = 0.5 and P(B) = 0.5, that is 0.5 x 0.5 = 0.25, or 25%. It is also equal to 1 minus the chance that at least one happens.