Probability Calculator
Two events combined, and the odds over repeated trials.
Work out Probability. Two events combined, and the odds over repeated trials. Shows the working, not just the answer.
P(A and B)
25.00%
P(A or B) is 75.00%
Assuming independence, P(A and B) is the product: 50% × 50% = 25.00%. That assumption is where most probability errors live — two events that feel unrelated often share a cause, and multiplying then understates the joint chance badly. Notice P(A or B) subtracts the overlap; adding the two probabilities double-counts it.
How the Probability Calculator works
Two events, combined every way that matters: both, either, exactly one, neither, and the conditional probability of one given the other. There is also the repeated-trials case, which answers the question people get most wrong — a one percent risk repeated a hundred times is a 63% chance of at least one occurrence, not one percent and not a certainty.
Also known as: probability of A and B · probability of at least one · conditional probability calculator · chance of something happening calculator
The calculation itself
For two independent events, P(A and B) is the product. P(A or B) is P(A) + P(B) − P(A and B), and that subtraction is not optional — simply adding the two double-counts the cases where both occur.
P(A given B) is P(A and B) ÷ P(B). When it equals P(A), the events are independent; when it differs, knowing B told you something about A.
In practice
Two fair coins: both heads is 0.25, at least one head is 0.75, exactly one is 0.50 and neither is 0.25. Those four outcomes partition the space and sum to exactly 1, which is a useful check on any probability calculation.
The repeated-trials case is the one that changes decisions. A 1% risk repeated 100 times gives a 63.4% chance of at least one occurrence — not 1%, and not the certainty the expected count of 1.0 might suggest.
Rolling a die four times gives a 51.8% chance of at least one six. That is the calculation behind one of the oldest results in probability, and it is close enough to even that intuition is no help.
Independence is the assumption that breaks
Multiplying two probabilities requires that one event tells you nothing about the other. This assumption is made constantly and holds less often than people think.
The failure mode is systematic and always in the same direction: events that feel unrelated frequently share a hidden cause, and the product then understates the joint probability. Two loans defaulting, two servers failing, two components wearing out — each of these looks independent until a recession, a power supply or a production batch turns out to be common to both.
That is why this page accepts the joint probability directly as an alternative. If you know P(A and B) from data rather than from an assumption, use it, and the conditional probability it produces will tell you how far from independent the events actually are.
A combination the page refuses
If P(A) is 50% and you enter a joint probability of 60%, the page returns nothing and explains why rather than printing a figure.
The reason is that 'both happened' is a subset of 'A happened'. The joint probability can never exceed either individual probability, and a calculator that silently accepts it would be producing arithmetic on an impossible state of the world.
Frequently asked questions
How do I calculate the probability of A and B?
For independent events, multiply them: two fair coins both landing heads is 0.5 × 0.5 = 0.25. If they are not independent the product is wrong, and you need the joint probability directly — which this page accepts as an input.
How do I calculate the probability of A or B?
Add them, then subtract the overlap: P(A) + P(B) − P(A and B). The subtraction matters — simply adding two probabilities double-counts the cases where both happen, and can produce a figure above 100%.
What is the chance of something happening at least once?
1 − (1 − p)ⁿ. A 1% chance over 100 trials is 63.4%, not 100% and not 1%. This is the calculation behind most 'it will not happen to me' errors.
What does independence mean here?
That one event happening tells you nothing about the other. It is assumed far more often than it holds — events that feel unrelated frequently share a cause, and multiplying their probabilities then understates the joint chance badly.
What is conditional probability?
P(A given B) is the chance of A once you know B happened: P(A and B) ÷ P(B). If it equals P(A), the events are independent. If it is higher, B makes A more likely.
Why can't the probability of both exceed either one?
Because 'both happened' is a subset of 'A happened'. If A occurs half the time, A-and-B cannot occur 60% of the time. This page rejects that combination rather than returning a meaningless figure.
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