Probability Calculator

Independent and mutually exclusive are not the same

Independent means one event does not affect the other's probability, so you multiply: two fair coins both landing heads is 0.5 × 0.5 = 0.25. Mutually exclusive means they cannot both happen, so you add: rolling a 1 or a 6 is 1/6 + 1/6 = 1/3. Confusing them is the most frequent error in probability, and the two conditions are close to opposites — mutually exclusive events are strongly dependent, since knowing one happened tells you the other did not.

The complement is often the easy route

The probability of at least one occurrence is almost always calculated as 1 minus the probability of none. "At least one six in four rolls" means computing (5/6)⁴ and subtracting from 1, giving about 0.518. Attacking it directly requires summing the cases for exactly one, two, three and four sixes, which is more work and more error-prone.

Conditional probability and the direction trap

P(A given B) and P(B given A) are different quantities, and swapping them is a mistake with real consequences. The probability that someone tests positive given they have a disease is not the probability they have the disease given they tested positive. Bayes' theorem converts between them, and the answer depends heavily on the base rate.

The base rate example worth knowing

A test 99 percent accurate for a condition affecting 1 in 10,000 people. Someone tests positive. Out of a million people, 100 have the condition and about 99 test positive; the 999,900 without it produce about 9,999 false positives. So the chance of actually having it is roughly 99 in 10,098 — under 1 percent. This result is unintuitive to nearly everyone, including clinicians in published studies, and it is the single most important idea in applied probability.

Independent events have no memory

A fair coin that has landed heads ten times has exactly a 0.5 chance of heads next. The gambler's fallacy is believing otherwise. Its inverse, the hot-hand belief that a streak makes continuation more likely, is the same error pointing the other way. Both come from expecting short sequences to look like long-run averages, which they do not.

Birthdays and why coincidence is common

In a room of 23 people, the probability that two share a birthday is over 50 percent, and at 70 people it exceeds 99.9. The count of pairs grows quadratically, so 23 people make 253 comparisons. This is why unlikely-looking coincidences occur constantly, and why hash collisions arrive far sooner than the size of the output space suggests.