Probability answers one question: out of everything that could happen, how much of it is the thing you want? A coin has two equally likely faces, so heads takes one share out of two. A die has six faces, so rolling a six takes one share out of six. Everything in probability is that same idea, made precise — count what counts as a success, count everything that could happen, and divide. The rules in this guide tell you how to combine those shares when two events are involved, when you want at least one success, and when the same trial repeats. No rule here needs anything beyond arithmetic and a clear statement of what you are counting.

What probability means

Probability is a number between 0 and 1 that measures how likely an event is. A probability of 0 means the event cannot happen. A probability of 1 means it is certain. A probability of 0.5 means it happens about half the time — the coin-flip case. The same chance can be written four ways, and fluency between them is half the subject: as a fraction (1 in 4 is 1/4), as a decimal (0.25), as a percentage (25%), and as odds (1 to 3 in favour — the odds dialect has its own guide, linked at the end of this article).

Two facts keep every calculation honest. First, the probabilities of all the possible outcomes of one trial add to 1. If a die shows 1, 2, 3, 4, 5 or 6, each with probability 1/6, the six shares add to 6/6, which is 1 — nothing is left over, and nothing is double-counted. Second, a probability is a statement about the setup, not a promise about the next trial. A fair coin that has landed heads five times in a row is still 50% to land heads on the sixth flip. The coin has no memory, and past flips do not change the shares.

The table above is worth memorising in outline, because so many questions reduce to recognising one of these shares. Notice that 1 in 4 and 25% are the same fact, and that neither is the same as odds of 1 to 3 — odds compare successes with failures, while probability compares successes with everything. Confusing the two is the most common source of wrong answers in this subject.

The probability formula, with worked examples

The probability formula applies whenever every outcome is equally likely — a fair coin, a fair die, a well-shuffled deck, a bag of identical marbles. It says: probability equals the number of favourable outcomes divided by the total number of possible outcomes. Favourable means the outcomes you are counting as a success, not outcomes that are somehow lucky. The formula is one division, and the worked examples below show the whole method: name the success, count it, count everything, divide, and simplify.

Start with a coin. The possible outcomes are heads and tails: two in total. The favourable outcome for heads is one. So the probability of heads is 1 divided by 2, which is 0.5, or 50%. A die is the same pattern with six outcomes. The probability of rolling a six is 1 divided by 6, about 0.1667, or 16.67%. The probability of rolling an even number is 3 divided by 6 — the faces 2, 4 and 6 — which simplifies to 1/2, or 50%.

Cards and marbles test whether you counted carefully. A standard deck has 52 cards, of which 4 are aces, so the probability of drawing an ace is 4 divided by 52, which simplifies to 1/13, about 7.69%. Half the deck, 26 cards, is red, so the probability of drawing a red card is 26 divided by 52, exactly 50%. A bag with 3 red marbles and 9 blue marbles has 12 marbles in total, so the probability of drawing red is 3 divided by 12, which is 1/4, or 25%. In each case the arithmetic is trivial; the work is in the counting, and the counting is where mistakes happen. Count the successes twice if the answer matters.

When the outcomes are not equally likely, this formula does not apply directly, and you need a different starting point — measured frequencies, or a model such as the bell curve. A weighted die, a biased coin, or tomorrow’s weather cannot be counted this way. The formula’s power comes from its narrowness: where the equal-likelihood assumption holds, the answer is exact, and where it does not, the formula itself tells you to stop and find the real shares another way.

The complement rule: everything the event is not

The complement of an event is everything that is not that event. If the event is rolling a six, its complement is rolling anything else: 1, 2, 3, 4 or 5. The complement rule says the two probabilities add to 1, so the probability of the complement is 1 minus the probability of the event. A 25% chance has a 75% complement. A 1-in-6 chance has a 5-in-6 complement. The rule looks too simple to matter, and it is the most useful rule in the subject.

Its power shows in at-least-one questions. The chance of getting at least one six in four rolls of a die is awkward to count directly — one six, two sixes, three sixes, four sixes — but its complement is a single case: no six at all. The chance of not rolling a six on one roll is 5/6. On four independent rolls it is (5/6) multiplied by itself four times, which is 625/1296, about 48.23%. Subtract from 1 and the chance of at least one six is about 51.77%. One subtraction replaced four separate counts, and the same trick answers every at-least-one question: work out the chance that it never happens, then flip it.

The same idea runs the repeated-trials calculation on the Probability Calculator: a 10% chance tried 10 times lands at least once 65.13% of the time, because the chance of failing all ten times is 0.9 multiplied by itself ten times, about 34.87%, and 100% minus 34.87% is 65.13%. Whenever a question says at least one, reach for the complement first.

The AND rule: both events happening

When a question asks for two things happening together — heads on both coins, a six and then another six — you are asking for an AND probability. For independent events, events where one does not change the other’s chances, the rule is to multiply: the probability of A and B equals the probability of A times the probability of B. Two coin flips are independent, so the chance of two heads is 1/2 times 1/2, which is 1/4, or 25%. The answer is smaller than either event alone, and that is correct: requiring both things to go right is harder than requiring one.

The hub’s default example shows the rule at work with percentages. With the probability of A at 30% and the probability of B at 40%, the chance of both is 30% times 40%, which is 12%. The Venn diagram on the calculator page shows the same 12% as the overlap of the two circles. AND always shrinks the probability when the events are independent, because each extra condition removes outcomes.

Independence is an assumption, and it must be checked, not presumed. Coin flips and die rolls are independent. Drawing cards without putting them back is not: the first draw changes what remains, so the second draw’s chances are different, and the multiply rule needs the updated shares — see the without-replacement section below. When two events influence each other but can still happen together, the general rule uses the conditional probability: the probability of A and B equals the probability of B times the probability of A given that B has happened. If you know the probability of A once B is known, that version of the rule always works, independent or not.

The OR rule: at least one of the events happening

When a question asks for at least one of two events — a six or an even number, rain on Saturday or rain on Sunday — you are asking for an OR probability. The rule is to add the two probabilities, then subtract the overlap you counted twice: the probability of A or B equals the probability of A plus the probability of B minus the probability of A and B. The subtraction is the whole point. The outcomes where both events happen were counted in A’s share and again in B’s share, so they must be removed once.

With the hub’s defaults, A at 30% and B at 40% with both at 12%, the OR probability is 30% plus 40% minus 12%, which is 58%. The remaining regions follow by the same bookkeeping: A only is 30% minus 12%, which is 18%; B only is 40% minus 12%, which is 28%; exactly one of the two is 18% plus 28%, which is 46%; and neither is 100% minus 58%, which is 42%. All four regions — both, A only, B only, neither — add to exactly 100%, which is a useful check on any answer.

Mutually exclusive events are the special case where the overlap is zero: the events cannot happen together at all. One card cannot be both a king and a queen, and one die roll cannot show both 3 and 5. For such events the OR rule simplifies to a clean add with nothing to subtract. The trap is treating exclusive events as independent, which invents an overlap that does not exist, or treating independent events as exclusive, which erases an overlap that does. Ask first: can both happen together? If yes, subtract the overlap. If no, add directly.

Without replacement: when the shares change

Drawing without replacement breaks the multiply rule in its simple form, because every draw changes the deck, the bag, or the batch. The first draw uses the full counts; the second draw uses the counts that remain. A bag with 5 red marbles and 7 blue marbles has 12 marbles. The chance the first draw is red is 5/12. If it is red, 4 red and 7 blue remain out of 11, so the chance the second is also red is 4/11. Multiply the updated shares: 5/12 times 4/11 is 20/132, which simplifies to 5/33, about 15.15%. The naive answer, 5/12 times 5/12, would be wrong, because it pretends the marble went back.

Counting combinations gives the same answer in one division and scales to bigger draws. The number of ways to choose k successes from K available, times the ways to choose the rest from the failures, divided by the ways to choose the whole draw from the population, is the hypergeometric formula — the exact distribution for sampling without replacement. The calculator page works an example: drawing 5 cards from a 52-card deck with 4 aces in it, the chance of exactly 2 aces is the combinations C(4, 2) times C(48, 3), divided by C(52, 5), which is 6 times 17,296 divided by 2,598,960, about 3.99%. The formula looks heavy and the idea is light: count the hands that work, count all the hands, divide.

With replacement, by contrast, the shares reset after every draw and the simple multiply rule holds — separate coin flips, die rolls, and draws where the card or marble goes back each time. The question to ask before any multi-draw calculation is therefore a practical one: does anything go back? The answer decides which rule you use, and getting it wrong is the difference between an exact answer and a plausible-looking wrong one.

Repeated trials: the binomial formula

When the same trial repeats with the same chance each time — five coin flips, four free throws, ten lottery tickets — the binomial formula answers exactly-k questions: exactly how many successes, out of how many trials? The formula has three parts: the number of orders in which the successes can arrive, the chance of the successes themselves, and the chance of the failures. Written out, the probability of exactly k successes in n trials with success chance p is C(n, k) times p raised to k, times (1 minus p) raised to (n minus k). The combination C(n, k) counts the orders; the powers supply the chance of one particular order.

Five fair coin flips make the arithmetic transparent. The chance of exactly 3 heads is C(5, 3), which is 10, times (1/2) raised to 5, which is 1/32 — giving 10/32, or 31.25%. Exactly 2 heads gives the same 31.25%, because C(5, 2) is also 10. At least 3 heads adds the exactly-3, exactly-4 and exactly-5 cases: 10 plus 5 plus 1, over 32, which is 16/32, or 50% — as symmetry demands, since heads and tails are interchangeable. You can check the arithmetic on the Probability Calculator’s repeated-trials tool, which reports the at-least-one, exactly-k and every-trial answers together.

With an uneven chance the same formula still works. A basketball player who makes 70% of free throws takes 4 shots. The chance of exactly 3 makes is C(4, 3), which is 4, times 0.7 cubed, which is 0.343, times 0.3, which is 0.1029 — giving 0.4116, or 41.16%. The formula needs independent trials with a fixed chance: if the chance drifts, or one trial changes the next, the binomial answer is an approximation at best, and the without-replacement counting above is the honest method instead.

Mistakes to avoid

Three mistakes account for most wrong probability answers. The first is adding where you should multiply, or multiplying where you should add. AND questions multiply (for independent events); OR questions add and subtract the overlap. If your answer to an AND question is bigger than either event alone, or your answer to an OR question is smaller than either event alone, the rule was applied backwards.

The second is the gambler’s fallacy: the belief that a run of one outcome makes the other outcome due. After five heads, the sixth flip is still 50% heads. Randomness has no memory and no sense of fairness; short runs of one outcome are normal, and the long-run averages the law of large numbers promises need very large numbers to show up. The third is confusing probability with odds. A 25% probability is odds of 1 to 3 in favour, not odds of 1 to 4 — odds compare wins with losses, and the conversion runs through the formulas in the odds guide. Keep the dialects separate and most answers check themselves: every probability between 0 and 1, every set of outcomes adding to 1, every AND answer smaller than its parts.

Frequently asked questions

What is the probability formula?

The probability formula is P = favourable outcomes ÷ total equally likely outcomes. Count the outcomes that count as a success, count all the outcomes that could happen, and divide. Three winning outcomes out of 12 gives 3 ÷ 12 = 1/4 = 25%. The formula needs equally likely outcomes — a fair coin, die, deck, or bag — to give an exact answer.

How do you calculate the probability of two events both happening?

If the events are independent, multiply: P(A and B) = P(A) × P(B). With P(A) = 30% and P(B) = 40%, the chance of both is 0.30 × 0.40 = 0.12, or 12%. If the events influence each other, use the conditional form instead: P(A and B) = P(B) × P(A given B), with the updated chance after B is known.

How do you calculate the probability of at least one event happening?

Use the complement: P(at least one) = 1 − P(none). For two events, P(A or B) = P(A) + P(B) − P(A and B) = 30% + 40% − 12% = 58%. For repeated trials, P(at least one) = 1 − (1 − p) raised to the number of trials — so a 10% chance tried 10 times lands at least once 65.13% of the time.

What is the difference between independent and mutually exclusive events?

Independent events do not affect each other — coin flips, die rolls — and they can happen together, with P(A and B) found by multiplying. Mutually exclusive events cannot happen together at all — one card cannot be both a king and a queen — so P(A and B) is 0 and P(A or B) is a clean add. Treating one kind as the other is the most common source of wrong answers.

How do you calculate probability without replacement?

Update the shares after every draw, because the population shrinks. Drawing two red marbles from 5 red and 7 blue gives 5/12 × 4/11 = 5/33, about 15.15%. For larger draws, count combinations: C(K, k) × C(N − K, n − k) ÷ C(N, n). Drawing 5 cards from a full deck, the chance of exactly 2 aces is about 3.99%.

What is the binomial probability formula?

The probability of exactly k successes in n independent trials with success chance p is C(n, k) × p raised to k × (1 − p) raised to (n − k). For exactly 3 heads in 5 fair coin flips: C(5, 3) = 10, times (1/2) raised to 5, giving 10/32 = 31.25%. The trials must be independent with a fixed chance for the formula to be exact.