Multiplication Rule: Dependent Events
The joint probability of two events A and B which are dependent is equal to the probability of A multiplied by the probability of B given that A has occurred.
P(A and B) =P(A) P(B|A)
or P(B and A) =P(B) . P(A|B)
This formula is derived from the formula of conditional probability of dependent events.
P(B|A) =
Þ P(A and B) = P(B|A).P(A)
Example
A study of an insurance company shows that the probability of an employee being absent on any given day P(A) is 0.1. Given that an employee is absent, the probability of that employee being absent a second day in succession P(B|A) is 0.4. Events A and B are dependent events because B cannot occur unless event A has occurred. The probability of an employee being absent on two successive days
P(A and B) =P(A) . P(B|A)
=(0.1) (0.4) = 0.04
Thus, the probability of an employee being absent on two successive days is 0.04 or 4% of the time.
Joint probability of several dependent events is equal to the product of the probabilities of occurrence of the preceding outcomes in the sequence.
P(A and B and C...) = P(A) P(B|A) P(C|A and B) ....