P(A, B)= P(A B) P(B) or P (A, B)=P(B A)P(A)
P(A|B)P(B) = P(B|A)P(A)
The Law of Total Probability:
P(B) = P(B/A).P(A) + P(B/A′) . P(A′)
Total probability and multiplication rule:
P(A/B) = P(B/A).P(A) (multiplication rule) P(B/A).P(A) + P(B/A′) . P(A′)(Law of Total Prob.) Where:
P(A): probability of occurrence of event A (marginal)
P(B): probability of occurrence of event B (marginal)
P(A′) = probability that A does not occur
P(B/A′) = probability that event B occurs given that A has not occurred already. P(A,B): Probability of simultaneous occurrence of events A and B (joint) P(A|B): Probability of occurrence of A given that B has occurred (conditional) P(B|A): Probability of occurrence of B given that A has occurred (conditional).
2. A continuous random variable is a random variable where the data can take infinitely many values. For example, a random variable measuring the time taken for something to be done is continuous since there are an infinite number of possible times that can be taken.
For any continuous random variable with probability density function f(x), we have that: