General Rules of Probability

General addition rule for any two events

For any two events A and B, P(A or B) = P(A) + P(B) - P(A and B)

Baye's rule

P(A|B) = P(B|A)P(A)/P(B)

When P(A) > 0, the conditional probability of B given A is

P(BIA) = P(A and B)/P(A)

Two events A and B are independent if

P(BIA) = P(B)

Bayesian statistics (Baye's rule)

Suppose B1, B2, .. Bn are disjoint events with positive probabilities, and the sum of these probabilities is 1.

Multiplication rule for any two events

The probability that two events A and B happen together can be found by: P (A and B) = P(A)P(BIA)

Two events are disjoint if

They can not occur at the same time (nothing in common), P(A and B) = 0

Tree diagram

A diagram used to show the total number of possible outcomes

Disjoint event

Events that have no outcomes in common

General addition rule for disjoint events

If A and B are disjoint events, P(A or B) = P(A) + P(B)

Disjoint vs independent

If A and B are disjoint, then the fact that A occurs tells us that B cannot occur - very dependent! Special multiplication rule holds if A and B are independent, but not otherwise

Multiplication rule for independent events

If two events A and B do not influence each other, the events are independent of each other. P(A and B) = P(A)P(B)

Two events are independent if

Knowing that one of the events has already occurred does not change the probability of the other event occurring.. P(AIB) = P(A)

Disjoint events are always associated with

The general addition rule of probability

Independent events are always associated with

The general multiplication rule of probability

Conditional probability

When we are trying to find the probability that one event will happen, given the info that the other event is already known to have occured