Probability
Probability quantifies uncertainty.
Probability is between 0 and 1.
P(A): Probability that the event A will occur.
- 0 <= P(A) <= 1
P(${A'}$): Probability that the event A will NOT occur.
- $0 <= P({A^c}) <= 1$
- $P(A) + P({A^c}) = 1$
- $P({A^c}) = 1 - P(A)$
Joint Probability
Define probability of two events occuring at the same time. ${P(A \cap B}) = P(A, B)$
If events A and B are true, then ${P(A,B) = P(A) * P(B)}$
An example of two independent events: A coin toss and a dice roll. A coin toss and dice roll are independent because the coin toss does NOT influence the outcome of the dice roll.
Probability of Union of Two Events
Define probability of two events as follows: ${P(A \cup B) = P(A) + P(B) - P(A \cap B)}$
Conditional Probability
Probability of an event given by another event. We define conditional probability of event B, given event A as follows:
- ${P(B | A) = P(B,A)/P(A)}$ if $P(A) > 0$
- ${P(A | B) = P(A,B)/P(B)}$ if $P(B) > 0$
Independence of Events
We say A is independent of B and B is independent of A iff: ${P(A|B) = P(A) * P(B)}$
This is called the unconditional independence.
Conditional Independence
We say that A and B are conditionally independent iff: ${P(A, B|C) = P(A|C) * P(B|C)}$
This essentially means that the dependence between A and B can be explained along with event C.
Difference between Independence and Mutual Exclusion
If events A and B are independent, the outcome of A does not affect the outcome of B, and vice versa.
If events A and B are mutually exclusive, then A and B cannot both be true ${P(A \cap B) = 0}$