actuary notes[2]

event

1. let A={x∣x被2整除，x∈N+}A=\{x| x被2整除，x \in \mathbb{N^+}\}A={x∣x被2整除，x∈N+},A′={x∣x不能被2整除，x∈N+}A'=\{x| x不能被2整除，x \in \mathbb{N^+}\}A′={x∣x不能被2整除，x∈N+},so $A^{\prime }$ can be called as the inverse (opposite) event of $A$.
2. at least one of the events $A$ and $A^{\prime }$ will certainly happen , so that is $A\cup A^{\prime }$.in the same way, the $A_1,A_2,.....$ events have one or more than one will definitely appear,that is $A_1\cup A_2\cup .....$ .
3. if both of $A$ and $A^{\prime }$ occur ,then that situation can be called as $A\cap A^{\prime }$.in a similar way, the $A_1\cap A_2\cap .....$ represents those event such as $A_1,A_2,.....$ all happen.
   for example, there are a great deal of corn kernels on a table,you take out some cof them for cooking, let A1={x∣x≥10}A_1=\{x|x \ge10\}A1​={x∣x≥10},A2={x∣x≤50}A_2=\{x|x \le 50\}A2​={x∣x≤50},A3={x∣x是偶数}A_3=\{x|x 是偶数\}A3​={x∣x是偶数},the $A^{\prime }=A_1\cap A_2\cap A_3=A_1{A}_2{A}_3$ reports the fact that the number of corn kernels you token from the table between 10 and 50 and will be divisible by 2.
4. as similar as sets,two events such as $A$ and $A^{\prime }$ have substraction operation that $A-A^{\prime }$,for example, let A1={x∣x≤30}A_1=\{x|x \le 30\}A1​={x∣x≤30},A2={x∣x≤50}A_2=\{x|x \le 50\}A2​={x∣x≤50},the A2−A1={x∣x≤50,x>30}A_2-A_1=\{x|x \le 50,x > 30\}A2​−A1​={x∣x≤50,x>30}.
5. if the two $A$ and $A^{\prime }$ events never happen concurrently, then they can be called as incompatible events,such as A={x∣x是偶数}A=\{x|x 是偶数\}A={x∣x是偶数} and A′={x∣x是奇数}A'=\{x|x 是奇数\}A′={x∣x是奇数}.
6. In probability theory, the operations on events (subsets of a sample space) follow specific algebraic rules similar to set theory. Here are the fundamental laws:

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1. Commutative Laws

• Union: $A\cup B=B\cup A$
• Intersection:$A\cap B=B\cap A$

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2. Associative Laws

• Union: $(A\cup B)\cup C=A\cup (B\cup C)$
• Intersection: $(A\cap B)\cap C=A\cap (B\cap C)$

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3. Distributive Laws

• Union over Intersection:
  $$A\cup (B\cap C)=(A\cup B)\cap (A\cup C)$$
• Intersection over Union:
  $$A\cap (B\cup C)=(A\cap B)\cup (A\cap C)$$

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4. De Morgan’s Laws (Duality Laws)

• Complement of Union:
  $$(A\cup B{)}^c=A^c\cap B^c$$
• Complement of Intersection:
  $$(A\cap B{)}^c=A^c\cup B^c$$

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5. Idempotent Laws

• Union: $A\cup A=A$
• Intersection: $A\cap A=A$

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6. Absorption Laws

• Union Absorption: $A\cup (A\cap B)=A$
• Intersection Absorption: $A\cap (A\cup B)=A$

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7. Complement Laws

• Double Negation: $(A^c{)}^c=A$
• Universal & Empty Set:
  $$S^c=\varnothing ,{\varnothing }^c=S$$
• Union with Universal Set: $A\cup S=S$
• Intersection with Empty Set: $A\cap \varnothing =\varnothing$

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8. Other Properties

• Set Difference:
  $$A\setminus B=A\cap B^c$$
• Symmetric Difference:
  $$A\Delta B=(A\setminus B)\cup (B\setminus A)$$

references

1. 《数学》