The complementAc of an eventA contains all outcomes that are not in A. The union {A or B} of events A and B contains all outcomes in A, in B, and in both A and B. The intersection {A and B} contains all outcomes that are in both A and B, but not outcomes in A alone or B alone.
The conditional probabilityP(B | A) of an event B, given an event A, is defined by P(B | A)=P(A and B)P(A) when P(A)>0. In practice, conditional probabilities are most often found from directly available information.
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The essential general rules of elementary probability are
Legitimate values:0≤P(A)≤1 for any event A
Total probability 1:P(S)=1
Complement rule:P(Ac)=1−P(A)
Addition rule:P(AorB)=P(A)+P(B)−P(AandB)
Multiplication rule:P(AandB)=P(A)P(B | A)
If A and B are disjoint, then P(AandB)=0. The general addition rule for unions then becomes the special addition rule, P(AorB)=P(A)+P(B).
A and B are independent when P(B | A)=P(B). The multiplication rule for intersections then becomes P(AandB)=P(A)P(B).
In problems with several stages, draw a tree diagram to organize use of the multiplication and addition rules.
If A1,A2,…,Ak are disjoint events whose probabilities are not 0 and add to exactly 1 and if B is any other event whose probability is not 0 or 1, then Bayes's rule can be used to calculate P(Ai | B) as follows: P(Ai|B)=P(B | Ai)P(Ai)P(B | A1)P(A1)+P(B | A2)P(A2)+⋯+P(B | Ak)P(Ak)