Conditional probability is the probability of an event given that another event has already occurred. It is found by restricting the sample space or using the formula P(A|B) = P(A∩B) / P(B).
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Key facts to remember
1P(A|B) means "the probability of A given B has occurred".
2P(A|B) = P(A ∩ B) / P(B).
3In practice: reduce the sample space to only the outcomes where B has occurred.
4Conditional probability changes the denominator.
5Without replacement changes probabilities — the second draw depends on the first.
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Formulas
Conditional probability
P(A|B) = P(A ∩ B) / P(B)
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Worked examples
Example 1
A bag contains 3 red and 5 blue balls. Two balls are drawn without replacement. Find P(second is red | first is red).
Working
After removing a red ball: 2 red remain, 5 blue remain, total = 7
P(second is red | first is red) = 2/7
Answer2/7
Example 2
In a class of 30: 12 study French, 10 study German, 4 study both. Find P(studies German | studies French).