　　Probability formula is an important link in probability calculation. Full probability formula, Bayesian formula, etc. can be applied to the probability of complex events, and all these formulas are derived from basic formulas.

Basic formula

　　For arbitrary events A and B　

　　Equation 1 says that the probability that A occurs is equal to 1 minus the probability that A does not occur (the probability of the opposite event). Another way of saying it may be better to understand, the probability of A happening plus the probability of A not happening equals 1, that is, event A either happens or doesn't happen. This is nonsense and an important truth, because it is often difficult to directly calculate the probability of A happening, but it is really easy to calculate the probability of A not happening.

　　Equation 2 says the probability that A occurs and B does not occur; Equation 3 is the probability that at least one of A and B occurs; Equation 4 does not explain. 2 and 3 are consistent with the set operation, and there are two equivalent ways of writing formula 3:

　　It's worth noting that you can't write something like this:

　　A and B are events, and events are sets, so the set symbol can be used, but P(A) and P(B) are probabilities, which are specific values, so the set symbol cannot be used.

mutually exclusive events

　　If there are two events A and B, if A occurs, B will not occur, and if B occurs, A will not occur, then they are mutually exclusive. If expressed as a set, then A∩B = φ . Mutually exclusive events are also called mutually exclusive events.

　　For mutually exclusive events A and B:

　　This actually comes from the basic formula:

_{For more}_mutually exclusive 　　events, if A 1 _,_A 2 _, A ₃ .

　　A more professional way of writing the above formula:

independent event

what is independence

　　Two events are independent, which intuitively means: in an experiment, the occurrence of one event does not affect the probability of the occurrence of the other event, and there is no relationship between the two. For example, the event that the dice rolls a "6" and the event that the dice rolls a "1" are independent of each other.

　　It should be noted that "mutual exclusion" describes a set relationship, and "independent" describes a probability relationship. The two are not in the same dimension, so don't try to link the two together.

　　There is a necessary and sufficient condition for independent events: if n events are independent of each other, then if any part of them is replaced by their respective opposite events, the resulting new n events are independent of each other:

official

　　For independent events A and B, the probability of both happening at the same time is equal to the product of the two:

　　Its equivalent written:

　　Note that the above formula holds only if A and B are independent events.

　　Extending to more independent events, if A ₁ , A ₂ , A ₃ ... A _n are independent of each other, then the probability of A ₁ ~ A _{n occurring simultaneously:}

　　More professional writing:

　　If A ₁ , A ₂ , A ₃ . . . A _n are independent of each other, then the probability that at least one of A ₁ ~ A _{n occurs if and only if:}

Conditional Probability

Hearing politics

　　Conditional probability refers to the probability that event B occurs under the condition that event A occurs, expressed in symbols:

　　The vertical line in the middle is regarded as a curtain, A is the queen mother, B is the young master, and A is under the curtain to B.

official

　　In fact, the two formulas are the same, and formula 2 is obtained by multiplying the left and right sides of formula 1 by P(A).

　　It should be noted that there is no indication that A and B are independent events. If A and B are independent events, according to the independent event formula, P(AB) = P(A)P(B), and the last item changes from P(B|A) to P(B), which means the occurrence of B It has nothing to do with A, that is, the queen mother wanted to listen to politics, but the young master grew up and did not listen to her.

　　Note that although P(AB) = P(BA), P(A|B)≠P(B|A):

　　Equations 1 and 2 can be used repeatedly for more events:

total probability formula

引例

　　一个村子里有三个独立作案的小偷，小偷参与盗窃的概率和盗窃能力已知，每次盗窃事件仅与其中一人有关，求村子失窃的概率。

　　有点难度了。

　　首先需要将问题转换成数学模型。令B事件为村子失窃事件，所求的是P(B)；设三个小偷A₁,A₂,A₃，小偷的全集就是 Ω = { A₁∪A₂∪A₃}；每次盗窃事件仅与其中一人有关，A₁,A₂,A₃是互斥的；小偷盗窃能力相当于该小偷在实施偷盗的情况下失窃的概率，P(B|A_i)。现在：

　　最终得到的就是全概率公式了，实际上就是由简单概率一步步推导而来，最重要的还是建立正确的概率模型。

公式

　　如果事件A₁、A₂、A₃…A_n 构成一个完备事件组，即它们两两互不相容，其和为全集Ω；并且P(A_i) > 0，则对任一试验B有：

　　右侧的两个表达式之所以相等，是因为：

不同的马甲

　　全概率公式的马甲众多，下面是教科书中的一个示例。

　　电子厂所用原件是由三个厂家提供的，已知以下数据：

原件场	次品率	供货份额
1	0.02	0.15
2	0.01	0.80
3	0.03	0.05

　　原件在仓库混合存放，每个元件没有明显区别，从仓库中随机取一个，次品的概率是多少？

　　数学模型：A₁,A₂,A₃分别表示元件是由三家元件场生产的，Ω = { A₁∪A₂∪A₃}；从仓库中随机取一个，得到次品的事件是B，P(B)为所求；P(A_i)是供货份额，P(B|A_i)是A_i的次品率。

贝叶斯公式

公式

　　大名鼎鼎的公式，常见的一个版本：

　　很多时候，求P(A|B)很困难，但求P(B|A)却很容易。上面的公式实际上是条件概率公式简单的推导：

　　结合全概率公式：

示例

　　对于上一节的元件次品问题，从仓库中随机取一个，如果是次品，那么该次品是哪个厂商的概率最大？

　　数学模型：A₁,A₂,A₃分别表示原件是由三家元件场生产的，从仓库中随机取一个，得到次品的事件是B，P(A_i|B)就表示次品是A_i生产的概率。

先下手为强

　　最后来看一个示例，甲乙二人轮流独立地对同一目标射击，谁先命中谁获胜，甲命中的概率是α，乙是β。现在由甲先射击，求二人获胜的概率分别是多少？

　　甲获胜的全集：{第1次射击时获胜∪第3次射击时获胜…∪第2n-1次射击时获胜}，n是自然数，n ≥ 1；甲第3次射击获胜的前提是，甲第1次射击失败且乙第2次射击失败，以此类推，甲的获胜全事件Ω = {甲第1次射击获胜∪(甲第1次射击失败∩乙第2次射击失败∩甲第3次射击获胜)…∪(甲第2n-1次射击时获胜(甲之前都失败∩乙之前都失败))}。用A、B分别表示甲乙的获胜事件，各事件之间相互独立（轮流独立射击），获胜事件之间互斥（谁先命中谁获胜，只能有一人胜出），下标表示二人出场次数，则：