## 统计代写|统计计算代写Statistical calculation代考|Probability rules for compound events

We can use various rules of probability to compute the probabilities of the more complex, related events.

1. If an event (E) is made up by two (or more) simple events ( $A$ and B), the probability $\mathrm{P}(\mathrm{E})$ can be formed either as:

the union of two or more events-all the outcomes that make up the two events

• the intersection of two events – the outcomes that fulfil the conditions for both events.
1. Two or more events are mutually exclusive if the occurrence of one event
2. means that none of the other events can occur at the same time. An outcome can belong to event $\mathrm{A}$ or to event $\mathrm{B}$ but not to both.
3. If you flip a coin, you can have either heads or tails. Both can’t happen at the same time!
4. Two events are independent if the occurrence of one is in no way affected by the occurrence of the other; that is, they are unrelated.
5. If you flip two coins and you obtained heads on the one, it will have no influence on the outcome of the second flip.
6. If there is a particular relationship between events such that the occurrence of one event affects the occurrence of the second event, the events are dependent. The probability attached to the occurrence of such events is known as conditional probability.

If the events are not mutually exclusive, it means that event A may occur or event B may occur, or both A and B may occur in a single outcome.

This rule states that the probability that either event $\mathrm{A}$ or event $\mathrm{B}$ occurs equals the probability that event A occurs plus the probability that event $B$ occurs minus the probability that both occur.
$$P(A \text { or } B)=P(A)+P(B)-P(A \text { and } B)$$
To avoid double counting the probability of the outcomes that fulfil the conditions for both events, $\mathrm{P}(\mathrm{A}$ and $\mathrm{B})$ is subtracted from the sum of the probability of $A$ and $B$.

Note: In this rule $\mathrm{P}(\mathrm{A}$ and $\mathrm{B})$ denotes the probability that $\mathrm{A}$ and $\mathrm{B}$ both occur in the same observation. In the multiplication rule $\mathrm{P}(\mathrm{A}$ and $\mathrm{B})$ denotes the probability that event A occurs on one trial followed by event B on another trial.

The probability that a person stopping at a petrol garage will ask to have his tyres checked is 0.12 , the probability that he will ask to have his oil checked is 0.29 and the probability that he will ask to have both checked is 0.07 . What is the probability that a person stopping at this garage will ask to have:

• either his tyres or his oil checked?
\begin{aligned} & \mathrm{P}(\mathrm{T} \text { or } \mathrm{O})=\mathrm{P}(\mathrm{T})+\mathrm{P}(\mathrm{O})-\mathrm{P}(\mathrm{T} \text { and } \mathrm{O}) \ & =0.12+0.29-0.07=0.34 \end{aligned}
• neither his tyres nor his oil checked?
$$1-\mathrm{P}(\mathrm{T} \text { or } \mathrm{O})=1-0.34=0.66$$

# 统计计算代写

## 统计代写|统计计算代写Statistical calculation代考|Probability rules for compound events

1. 如果事件 (E) 由两个（或更多）简单事件组成（A和 B), 概率P(和)可以形成为：

• 两个事件的交集——满足两个事件条件的结果。
1. 如果一个事件的发生，则两个或多个事件互斥
2. 意味着没有其他事件可以同时发生。结果可以属于事件A或参加活动乙但不是两者。
3. 如果你抛硬币，你可以得到正面或反面。两者不能同时发生！
4. 如果一个事件的发生绝不会受到另一个事件的影响，则两个事件是独立的；也就是说，它们是无关的。
5. 如果您掷两枚硬币，并且您在一枚硬币上正面朝上，则不会影响第二次掷硬币的结果。
6. 如果事件之间存在特定关系，使得一个事件的发生影响第二个事件的发生，则这些事件是相关的。发生此类事件的概率称为条件概率。

$$P(A \text { or } B)=P(A)+P(B)-P(A \text { and } B)$$

• 他的轮胎或机油检查了吗?
$$\mathrm{P}(\mathrm{T} \text { or } \mathrm{O})=\mathrm{P}(\mathrm{T})+\mathrm{P}(\mathrm{O})-\mathrm{P}(\mathrm{T} \text { and } \mathrm{O}) \quad=0.12+0.29-0.07=0.34$$
• 他的轮胎和机油都不检查吗?
$$1-\mathrm{P}(\mathrm{T} \text { or } \mathrm{O})=1-0.34=0.66$$

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