How this instrument works
When two events are independent — the outcome of one has no effect on the outcome of the other — the probability that both happen is the product of their individual probabilities: P(A and B) = P(A) x P(B). It's the multiplication rule, and it's why flipping two coins and getting heads on both is far less likely than getting heads on just one.
The everyday cases are the clearest ones: two separate coin flips, two separate die rolls, two unrelated customers each deciding whether to buy. Each event's chance stays exactly what it was on its own; multiplying just asks how often both of those independent things line up at once.
This rule only holds when the events genuinely don't affect each other. If knowing that A happened would change your estimate of B's probability — drawing a second card without replacing the first, say — the events are dependent, and straight multiplication overstates or understates the true joint probability. This site's conditional-probability calculator handles that dependent case correctly.
- Enter the first event's chance into P(A) (%) — as a percentage, like 50 for a coin flip.
- Enter the second event's chance into P(B) (%).
- Read P(A and B) (%) — independent events beneath them; it's simply P(A) times P(B), scaled back down to a percentage.
- This instrument assumes independence. If knowing A changes the chance of B — drawing cards without replacing the first — use this site's conditional-probability calculator instead.
- Both P(A) and P(B) are entered as percentages from 0 to 100, not decimals — enter 50, not 0.5, for a fair coin.
Worked example — two coin flips both landing heads
Set P(A) (%) to 50 and P(B) (%) to 50 — the chance of heads on two separate, independent coin flips. P(A and B) (%) — independent events reads 25%, since 50% x 50% = 25%. Direct enumeration confirms it: HH is exactly one of the four equally likely two-flip outcomes (HH, HT, TH, TT), and 1 of 4 is 25%.
Independence is doing real work in that calculation. If the two 50% events were dependent instead — like drawing two cards from a deck without replacing the first, where the second draw's odds shift once the first card is gone — this straight multiplication would give the wrong answer, and you'd need this site's conditional-probability calculator to combine them correctly.
Questions
What does 'independent events' mean here?
Two events are independent when knowing that one occurred tells you nothing new about the other's probability. Two separate coin flips are independent — the first flip's result doesn't change the odds of the second. Drawing two cards from a deck without replacement is not independent, because removing the first card changes what's left for the second.
Why do I multiply instead of add for 'and'?
Because you need both events to happen simultaneously, and each additional requirement narrows the pool of outcomes that qualify. Multiplying two probabilities each less than 100% always produces a smaller number — reflecting that 'both A and B' is a stricter, less likely condition than either one alone. Addition is the rule for 'at least one,' handled by this site's or-probability calculator.
What if my two events aren't actually independent?
Then P(A) x P(B) will give the wrong joint probability. For dependent events, you need P(A and B) = P(A) x P(B|A), where P(B|A) is B's probability given that A already happened. This site's conditional-probability calculator computes that adjusted, conditional figure directly.
How is this different from the joint-probability calculator?
They use the identical independent-events multiplication rule, P(A) x P(B). This calculator frames it around everyday one-off events like coin flips and dice. The joint-probability calculator applies the same formula in a statistics-textbook setting — reading two probabilities out of a joint distribution table, such as weather and traffic outcomes assumed independent — and is built for that table-based context.
Can P(A and B) ever be larger than P(A) or P(B) alone?
No, not for independent events. Multiplying two numbers that are each at most 100% never increases the result — P(A and B) is always less than or equal to the smaller of P(A) and P(B). Requiring two things to both be true is always at least as restrictive as requiring just one.
What happens if one event is impossible or certain?
If either P(A) or P(B) is 0%, P(A and B) is 0% — an impossible event can never co-occur with anything. If one of them is 100%, P(A and B) simply equals the other event's probability, since a certain event doesn't reduce the joint chance at all.