Probability in plain language
Probability is a number that describes uncertainty. Zero means an outcome cannot happen, one means it must happen, and values between them describe different degrees of likelihood. The same value can be written as a decimal, fraction, or percentage: 0.25, one quarter, and 25% all represent the same probability.
The arithmetic is usually short. The difficult part is deciding which relationship exists between the events. Are they independent? Can they overlap? Does knowing one occurred change the chance of the other? A calculator can apply a formula precisely, but the formula only answers the intended question when that relationship is identified correctly.
One event and its complement
For a single event A, enter its probability directly. Its complement means “not A” and covers every outcome outside A.
P(not A) = 1 − P(A)
If rain has a 30% probability, no rain has a 70% probability. The two must total 100% because exactly one of them will occur. Complements are often the quickest route to questions containing “at least one.” Instead of listing every way one or more successes can happen, calculate the probability of zero successes and subtract it from one.
For example, suppose a component has a 2% failure probability and three components fail independently. The chance that none fails is 0.98 cubed, approximately 94.12%. Therefore the chance that at least one fails is about 5.88%. Adding 2% three times gives 6%, which is close but not exact because it double-counts outcomes where several components fail.
Independent events: A and B
Events are independent when learning that one happened gives no information about the other. Separate fair coin tosses are the standard example. The second toss remains 50–50 regardless of the first result.
For independent events, multiply to find the probability that both occur:
P(A and B) = P(A) × P(B)
Rolling a six on a fair die has probability 1/6. Flipping heads has probability 1/2. Doing both is 1/6 × 1/2 = 1/12, about 8.33%.
Independence should never be assumed simply because two events have different names. Weather can affect both attendance and sales. A person’s age can affect both income and insurance costs. Drawing two cards without replacing the first makes the second draw depend on the first because the deck has changed.
Independent events: A or B
The word “or” in probability normally includes the possibility that both events happen. Adding P(A) and P(B) counts their shared outcomes twice, so the intersection must be subtracted once.
P(A or B) = P(A) + P(B) − P(A and B)
For independent events, P(A and B) is their product. With A at 40% and B at 30%, the chance of either is 40% + 30% − 12% = 58%.
The subtraction is not optional. A raw sum of 70% treats “A only,” “B only,” and “both” as separate even though the events where both occur already appeared inside each original percentage.
Mutually exclusive events
Mutually exclusive events cannot both occur in the same trial. A single die roll cannot be both two and five. Their intersection is zero, so their probabilities can be added directly.
P(A or B) = P(A) + P(B) when P(A and B) = 0
This differs sharply from independence. Independent events are allowed to happen together; mutually exclusive events are not. If two events each have a positive probability and are mutually exclusive, observing one makes the other impossible, so they are necessarily dependent.
Conditional probability
Conditional probability asks for the chance of A given that B is already known to have happened. The notation is P(A|B), read “probability of A given B.”
P(A|B) = P(A and B) ÷ P(B)
The denominator restricts attention to the B outcomes. The numerator counts the part of those outcomes that also belongs to A. If 30% of customers use mobile devices and 15% both use mobile and make a purchase, then the purchase probability among mobile customers is 15% ÷ 30% = 50%.
P(B) cannot be zero because conditioning on an impossible event leaves no outcome set to measure. The intersection also cannot exceed either original event: everything in “A and B” must belong to A and to B.
Converting percentages, decimals, and fractions
Divide a percentage by 100 to obtain a decimal. Multiply a decimal by 100 to obtain a percentage. Fractions divide numerator by denominator: 3/8 is 0.375 or 37.5%.
Keep full precision during a calculation and round only the displayed result. Repeated early rounding can noticeably distort chains of small probabilities. For a result used in safety, medicine, finance, or engineering, retain enough significant figures for the decision and document assumptions rather than relying only on a visually tidy percentage.
Common mistakes
The most common error is adding probabilities for overlapping events without subtracting the overlap. The second is multiplying events that are not independent. A third is mixing percentages and decimals: 40% is 0.40 in a formula, not 40. Entering 40 as a decimal probability would imply 4,000%, which is impossible.
Another mistake is confusing probability with odds. A probability of 75% means three successful outcomes for every one unsuccessful outcome, so the odds in favor are 3 to 1. It does not mean odds of 75 to 1. Probability compares success with all outcomes; odds compare success with failure.
Finally, a calculated probability inherits the quality of its inputs. Historical frequency may not predict a changing system, and an estimated 20% may hide substantial uncertainty. The answer is mathematically exact for the inputs while the real-world estimate can still be imprecise.
Worked examples
Two independent screening steps pass 90% and 80% of items. The probability an item passes both is 0.9 × 0.8 = 0.72, or 72%. The complement, failing at least one, is 28%.
An email campaign has a 45% open probability and a 10% click probability, with independence assumed only for illustration. The probability of an open or click is 0.45 + 0.10 − 0.045 = 0.505, or 50.5%. In practice clicks usually depend on opens, so real campaign analysis should use observed intersection data instead.
In a class, 60% study mathematics, 35% study computing, and 20% study both. The chance that a randomly selected student studies either is 60% + 35% − 20% = 75%. The chance a computing student also studies mathematics is 20% ÷ 35%, approximately 57.14%.
Reading the result responsibly
A probability does not predict what will happen in one trial. A 90% event can fail, and a 1% event can occur. Calibration appears across many comparable trials: events assigned roughly 70% probability should happen about seven times in ten over a sufficiently large, stable sample.
Small probabilities also need context. A one-in-a-million event may be negligible for one attempt but important across billions of attempts. Conversely, a large percentage attached to a harmless outcome may deserve little attention. Good decisions combine probability with impact, exposure, and the cost of reducing the risk.