Conditional probability — Mathematics, 14–17 years
How new information changes the chance of an event. Mathematics, 14–17 years.
The idea
A probability can change when you learn that some cases are no longer possible. “The chance of rain” is different from “the chance of rain given that the sky is already cloudy”. Conditional probability focuses only on the cases matching the information you have.
Why it matters
People often treat a clue as if it changed nothing, or as if it proved the answer. The real problem is to update a chance carefully without ignoring the cases ruled out by the clue. This idea grew from games of chance and now supports medicine, testing, insurance and decisions under uncertainty.
A worked example
A bag has 3 red and 2 blue counters. Two are drawn without replacement. Given that the first is red, 2 red and 2 blue remain, so the chance that the second is red is 2 out of 4, or 1/2. The condition changes the contents before the second draw.
The common trap
A natural mistake is to keep using 3/5 for the second draw, because that was the original chance of red. But learning that a red counter was removed leaves only four counters, including two red ones. The first draw is not independent of the second when the counter is not replaced.
Where it appears
A medical test result is interpreted together with symptoms, age and how common the illness is. A positive result can mean different things in a rare illness and a common one. Conditional probability helps compare the chance of illness after the result, rather than confusing it with the chance of a positive result when illness is present.
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