Probability answers a question that sounds trivial and is not: how likely is this, given what I know? The arithmetic is elementary. What takes practice is recognising which situation you are in, because two of the commonest mistakes come from choosing between adding and multiplying wrongly.
Probability as a ratio
For equally likely outcomes, the probability of an event is simply the count of favourable outcomes over the total:
P = favourable ÷ total
Roll a 3 on a fair six-sided die: 1 ÷ 6 ≈ 0.1667, or 16.67%, written as the fraction 1/6. All three notations are the same number, and the fraction is usually the exact one — the probability calculator returns the decimal, the percentage and the fraction simplified to lowest terms, because rounding a fraction is how small errors get introduced.
The complement rule is the workhorse
P(not E) = 1 − P(E)
"At least one" problems are the classic application, and they turn ugly fast if you enumerate directly. Six dice, at least one six: enumerating means summing six terms for exactly one six, fifteen for two, and so on. The complement is instant — P(no six) = (5/6)⁶ ≈ 0.3349, so P(at least one) = 1 − 0.3349 ≈ 0.665.
One line, and no possibility of a summation error. Reach for the complement whenever the direct route looks tedious, and it usually is.
Independence: multiply
For independent events, the probability of all of them is the product:
P(A and B) = P(A) × P(B)
Two fair coin flips: 0.5 × 0.5 = 0.25, not 0.5. This is where a lot of intuition goes wrong — each flip is independent of the others, and the second one does not become less likely because the first landed heads. The random number generator exists precisely because human beings are poor at judging randomness; the arithmetic of independence holds, our instinct for it does not.
More than two events just extends the chain: three flips give 0.125. The rule of thumb: the product shrinks fast, so "at least one of 10 independent things each with a 10% chance" is about 65%, not 10%.
Mutual exclusion: add
For mutually exclusive alternatives — rolling a 2 or a 3 on one die — you add, because any one can happen but two cannot happen together:
P(A or B) = P(A) + P(B)
1/6 + 1/6 = 1/3. Compare with rolling two dice and asking for "a 2 on either": there the events are independent, so the correct route is 1 − (5/6)² = 0.306, not 1/3.
Both the sums and the products are easy to mix up because both feel like combining "chances". The test: ask whether both can happen at once. If yes, multiply. If no, add.
Odds are not probability
Probability is favourable ÷ total. Odds in favour are favourable ÷ unfavourable — a ratio of two counts rather than a fraction of the whole.
Three successes in twelve trials: P = 3/12 = 25%, but the odds are 3 : 9, which is 1 to 3.
This trips people up because odds sound bigger than probabilities when they are smaller, and vice versa. Odds of 3 to 1 correspond to a probability of 3/4 = 75%, not 30%. Betting markets, actuarial tables and clinical risk all quote odds; everything else quotes probability. Know which convention is in front of you before comparing two numbers.
Permutations and combinations
When order does not matter but selection does, you want combinations:
n! ÷ k!(n−k)!
The number of ways to draw 3 cards from a 52-card deck is 22,100. When order matters, use permutations: 52 × 51 × 50 = 132,600. Same cards, different question — so the ratio between the two is not academic, it is the difference between "what is in the hand" and "what is the order".
Common mistakes
Adding when you should multiply. Two independent 50% events give 25%, not 100%.
Assuming independence when events are linked. Drawing a card and then drawing again from the same deck is without replacement — the events are dependent, and the naive product is wrong. This is the classic textbook trap and the reason the question always specifies "without replacement".
Rounding the decimal and multiplying. Keep fractions until the end. With many factors, small rounding errors compound, and 0.9 × 0.9 is 0.81, not 0.8.
Treating "at least one" as a simple count. Overlapping cases need the complement rule.
Where probability earns its keep
Beyond dice and cards: the chance of a system meeting a deadline is the complement of every task slipping. Insurance pricing is an expectation built from probabilities. Quality control flags an item as suspect once its probability of being defective falls below a threshold. A/B tests compare a conversion rate against the null hypothesis that the change did nothing. Medical risk communication — "1 in 10,000" versus "0.01%" — is a lesson in how the same number lands depending on its form.
For the spread of values around an average, which is the other half of the picture, see the standard deviation guide.
Conditional probability and why the base changes again
When the question changes — "given that this happened, how likely is that?" — the base shifts, and the formula changes with it:
P(A given B) = P(A and B) ÷ P(B)
This is the same asymmetry story that makes percentages lie, now applied to probabilities. A medical test with 99% accuracy for a condition affecting 1 in 10,000 people produces a positive result that is about 1% likely to be correct. The arithmetic is not in dispute; the surprise comes entirely from the base rate being tiny, which is exactly what a conditional probability exposes.
Bayes' theorem is this idea rearranged, and it is how evidence updates belief. A frequentist probability is about a single trial; a Bayesian one keeps updating as data arrives, which is why Bayesian methods dominate medical diagnostics, spam filtering and any system that learns.
Expected value: probability turned into a number
Multiply each outcome by its probability and add, and you get the expected value — the long-run average you would see over many repetitions.
A slot paying $10 with probability 1/5 and $0 otherwise has an expected value of $2. That does not mean you will win $2; it means across a very large number of plays you would average $2. The distinction is the whole point: expected value describes a distribution, not an outcome, and conflating the two is how people lose money on fair-looking games.
Expected value is also how insurance is priced. An insurer's expected payout per policy is priced plus a margin, and the difference between those two numbers is the entire business model.
Frequently asked questions
How do I calculate a probability?
Divide the number of favourable outcomes by the total number of equally likely outcomes: P = favourable ÷ total. Then express it as a fraction, decimal and percentage as needed.
What is the complement rule?
P(not E) = 1 − P(E). It is the shortcut for 'at least one' problems: compute the easier probability that nothing happens, then subtract from 1.
How do odds differ from probability?
Probability is favourable ÷ total; odds in favour are favourable ÷ unfavourable. Odds of 3 to 1 correspond to a probability of 75%, not 30%.
When do I multiply probabilities instead of adding them?
Multiply for independent sequential events — each coin flip, each die roll. Add only for mutually exclusive alternatives where any one can happen but two cannot both happen.