Probability Calculator: Two Events, Step by Step

All the outputs below assume independent events. Probabilities are written as decimals between 0 and 1; multiply by 100 for a percentage.

• P(A and B) — both occur: P(A) × P(B)
• P(A or B) — at least one occurs: P(A) + P(B) − P(A and B)
• P(A only): P(A) × (1 − P(B))
• P(B only): (1 − P(A)) × P(B)
• P(exactly one): P(A only) + P(B only), or equivalently P(A or B) − P(A and B)
• P(neither): 1 − P(A or B)
• Complements: P(not A) = 1 − P(A); P(not B) = 1 − P(B)
• P(A given B): for independent events, P(A given B) = P(A) — knowing B happened changes nothing

Flip a fair coin twice. Let A = heads on the first flip, B = heads on the second flip: P(A) = 0.5, P(B) = 0.5.

• P(A and B) = 0.5 × 0.5 = 0.25 — two heads in a row, 25%
• P(A or B) = 0.5 + 0.5 − 0.25 = 0.75 — at least one heads, 75%
• P(A only) = 0.5 × 0.5 = 0.25 — heads then tails
• P(neither) = 1 − 0.75 = 0.25 — two tails

The four outcomes — heads-heads, heads-tails, tails-heads, tails-tails — are equally likely at 0.25 each, and the four numbers above just group them.

Two inputs, eight answers. Decimals only — 0.5 means a 50% chance, 0.05 means 5%.

Step by step

1. Enter P(A). The probability of the first event, as a decimal between 0 and 1. For example, a 30% chance is 0.3.
2. Enter P(B). The probability of the second event, same format.
3. Press Calculate. Read P(A and B), P(A or B), the only-one/either-neither combinations, and the step-by-step working underneath.

Percentages and probabilities convert freely: just multiply a decimal by 100. A percentage calculator can handle the conversions if you think in percent.

The formulas change completely with the relationship between the events. This calculator handles the independent case, where one event does not affect the other.

For mutually exclusive events, P(A or B) is simply P(A) + P(B) — the subtraction term vanishes because both cannot happen. For dependent events you need conditional probability, P(A given B), which this calculator does not attempt.

These rules are not just textbook exercises:

• Games and lotteries. The chance of winning twice in a row, or of at least one win across several plays, follows exactly these formulas.
• Risk and insurance. The probability that at least one of two independent failures hits — two machines, two suppliers — drives redundancy planning.
• Medicine and testing. The chance that at least one of two independent tests catches a condition is P(A or B), which is why retesting helps.
• Everyday planning. If two independent events each have a 90% chance of going right, the chance both go right is 0.9 × 0.9 = 0.81 — only 81%, a useful reality check for optimistic plans.

Averages and rates feed into many of the same analyses; an average calculator covers the mean, median and mode side. For the mathematical foundations, Wikipedia's overview of probability is a solid, well-referenced starting point.

• Adding probabilities for overlapping events. P(A or B) = P(A) + P(B) only works when both cannot happen at once. Otherwise you double-count the overlap — that is what the − P(A and B) term fixes.
• The gambler's fallacy. After five tails, the next flip is still 0.5 heads. Independent events have no memory.
• Confusing P(A and B) with P(A given B). Both happening is P(A) × P(B); A given that B already happened is a conditional probability and needs different information.
• Percentages above 100. No probability exceeds 1 (or 100%). If your inputs combine to more than 1 somewhere unexpected, re-check whether the events are really independent.
• Assuming independence without checking. Real events are often subtly linked — weather and traffic, for example. The formulas are only as good as the assumption.