⋅ P(A and B): probability of event A occurring and then event B occurring in successive trials.
⋅ P(B | A): probability of event B occurring, given that event A has already occurred.
⋅ P(A and B) = P(A) ⋅ P(B | A)
⋅ Independent Events
If the events A, B are independent, then the knowledge that event A has occurred has no effect on the probably of the event B occurring, that is P(B | A) = P(B).
As a result, for independent events: P(A and B) = P(A) ⋅ P(B | A)
= P(A) ⋅ P(B)
Independent Events VS. Dependent Events
Bag A contains 2 red balls and 3 green balls. Bag B contains 1 red ball and 4 green balls.
A fair die is rolled: if a 1 or 2 comes up, a ball is randomly selected from Bag A;
if a 3, 4, 5, or 6 comes up, a ball is randomly selected from Bag B.
Multiplication rule for "AND"
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