Independent, exclusive, or neither
Independent means one event tells you nothing about the other — two separate coin flips. Then P(A and B) = P(A)×P(B). Mutually exclusive means they cannot both happen — a single die showing 3 and showing 5. Then P(A and B) = 0, and P(A or B) is simply the sum.
Most real pairs are neither: rain and a delayed train are related but not exclusive. For those, supply P(A and B) directly and everything else follows.
The 'at least once' trap
People consistently underestimate how likely a rare event becomes when repeated. A 10% chance run ten times is not a 100% chance — but it is not 10% either. It is 1 − 0.9¹⁰, about 65%. That single calculation explains most of why unlikely failures keep happening in systems that run continuously, and why it is worth designing for.
Frequently asked questions
Why did it reject my P(A and B)?
Because no pair of events can produce it. The joint probability can never exceed either individual probability, and it can never be below P(A)+P(B)−1. Values outside that window are mathematically impossible, not merely unlikely.
Is my data uploaded?
No. Everything is worked out in the page and nothing is stored.
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