Probability: a quantitative indicator to measure the probability of something happening, which can be used to measure the possibility of something happening. Probability = number of possible bets/number of all possible outcomes.

Event: indicates a result or event with probability.

Probability representation: P(A)=n(A)/n(S) indicates the probability of occurrence time A, n(A) indicates the possible number of occurrence events A, n(S) indicates the number of all possible results, and s is called probability space or sample space.

Opposite event: A occurs, a' means that the opposite event of A -A does not occur, and P(A')= 1-p(A).

Venn diagram: Draw a box to represent the sample space S, and draw several circles to represent related events. draw

Probability is only an expression of the possibility of an event, and probability is not a guarantee. An impossible event can still happen as long as it is not completely impossible.

Mutually exclusive events: If two events are mutually exclusive events, then only one of them will happen, P(A∩B)=0.

Cross events: If two events cross, these two times may occur at the same time.

Intersection: A∩B stands for the intersection of A and B. This symbol can be understood as "AND" to find the same element of different events.

And: A∪B stands for the union of A and B. This symbol can be understood as "or" and contains all elements belonging to A and B.

Exhaustive: If P(A∪B)= 1, then we say that A and B are exhaustive, and together they constitute the whole S, which exhausts all possibilities.

P(A∪B)=P(A)+P(B)？ P(A∩B)

P(A)=P(A∩B)+P(A∩B'), which seems complicated and unnecessary, but it is very convenient and practical in practical calculation.

Conditional probability: If event B is the probability of event A with known conditions, it can be abbreviated as P(A|B).

Probability tree

The probability of various results is marked on the first-level branch line, and the probability of the second-level result when the connected upper-level result is known is marked on the second-level branch line, and the second-level is the conditional probability.

Multiply a probability by the following branch probability, and you can find the probability that contains the intersection.

The sum of all probabilities of each level branch must be 1, as shown in the figure.

The purpose of venn diagram is to point out the basic probability and various relationships, and the purpose of probability number is to calculate the conditional probability.

Total probability theorem

Calculate the total probability of a specific event according to the conditional probability. It is known that P (a), P(B | a) and P (b' | a) can find P(B).

The total probability formula is the denominator of Bayesian theorem.

Thomas Bayes

Bayesian theorem provides a method to calculate the inverse conditional probability. It is known that P(B|A) can be found.

Related events: If

Then we say that event A and event B are related events, and the probabilities of event A and event B influence each other.

Independent event: If event A and event B are independent of each other, the probability of event A is not affected by event B.. Independent event

, so

Independent events and mutually exclusive events: If it is mutually exclusive events, it is not an independent event; If it was an independent event, it wouldn't be mutually exclusive events.