Conditional probability, total probability formula, Bayesian formula

1. Basic concepts

1. Conditional probability

Conditional probability is the probability that event A will occur given the occurrence of event B. Conditional probability is expressed as: P(A|B), read as "the probability of A occurring given the occurrence of B ". If there are only two events A, B, then,

$P(A|B)=\frac{P(A,B)}{P(B)}$

Extended formula:

$P(A,B)=P(A|B)\cdot P(B)$

$P(A,B|C)=P(A|B,C)\cdot P(B|C)$

2. Probability measure

If the probability P ( B ) of event B > 0, then the function Q defined by Q ( A ) = P ( A | B ) over all events A is a probability measure. If P ( B ) = 0, P ( A | B ) is undefined. Conditional probabilities can be calculated using decision trees

3. Joint probability

Indicates the probability of two events occurring together. The joint probability of A and B is expressed as P(AB) or P ( A , B ), or P(A∩B). [2]

4. Marginal probability

is the probability of an event occurring independent of other events. The marginal probability is obtained in this way: in the joint probability, those events that are not needed in the final result are combined into the total probability of its events and disappear (for discrete random variables, the total probability is obtained by summing, and for continuous random variables, the total probability is obtained by integral probability). This is called marginalization . The marginal probability of A is denoted as P ( A ), and the marginal probability of B is denoted as P ( B ).

2. Fundamental theorem

1. Theorem 1

Assuming that A and B are two events, and A is not an impossible event, it $P(B|A)=\frac{P(AB)}{P(A)}$ is called the conditional probability of event B occurring under the condition that event A occurs. normally,

$P(B|A)\n P(B)$ , and it satisfies the following three conditions:

(1) Non-negativity; (2) Normative; (3) Additivity.

2. Theorem 2

Let E be a random experiment, Ω be the sample space, A and B be any two events, let P(A)>0, $P(B|A)=\frac{P(AB)}{P(A)}$ which is called probability of event B under the condition of "event A occurs".

The above multiplication formula can be extended to the case of any finite number of events.

Set as $A_{1},A_{2},...,A_{n} are any n events$ any n events (n>=2) and $P(A_{1}A_{2}...A_{n})> 0$ , then

$P(A_{1}A_{2}...A_{n})=P(A_{1})P(A_{2}|A_{1})...P(A_{n}|A_{1}A_{2}...A_{n-1})$

3. Theorem 3 (full probability formula)

Definition: (partition of complete event group/sample space)

Let B1, B2,...Bn be a group of events, if

(1) $\forall i\neq j\in \left \{ 1,2,...,n \right \},B_{i}\bigcap B_{j}= \Phi$

(2) $B_{1}\bigcup B_{2}\bigcup ...\bigcup B_{n}=\Omega$

Then it is called a division of B1, B2, ... Bn sample space Ω, or a complete event group of sample space Ω.

Total probability formula:

Let the event group $\left \{ B_{i} \right \}$ be a partition of the sample space Ω, and P(Bi)>0 (i=1, 2,...n)

Then for any event B, we have $P(A)=\sum_{i=1}^{n}P(B_{i})P(A|B_{i})$

4. Theorem 4 (Bayesian formula)

Suppose B1, B2, ... Bn ... is a complete event group, then for any event A, P(A)>0, we have

$P(B_{i}|A)=\frac{P(B_{i}A)}{P(A)}=\frac{P(A|Bi)P(Bi)}{\sum_{i=1 }^{n}P(A|Bi)P(Bi)}$