Question

For any three events $A,B$ and $C$ we have

• $P$ (at least one of $A,B$ and $C$ occurs)
  
• $P$ (at least two of $A,B,C$ occur)
  
• $P$ (exactly two of $A,B,C$ occur)
  
• $P$ (exactly one of $A,B,C$ occurs)
  

Answer 1

1. $P$ (at least one of $A,B$ and $C$ occurs)= $P\left(A\right)+P\left(B\right)+P\left(C\right)-P(A\cap B)-P(B\cap C)-P(C\cap A)+P(A\cap B\cap C)$
   
2. $P$ (at least two of $A,B$ and $C$ occurs) $=P(B\cap C)+P(C\cap A)+P(A\cap B)-2P(A\cap B\cap C)$
   
3. $P$ (exactly two of $A,B$ and $C$ occurs) $=P(B\cap C)+P(C\cap A)+P(A\cap B)-3P(A\cap B\cap C)$
   
4. $P$ (exactly one of $A,B$ and $C$ occurs) $=P\left(A\right)+P\left(B\right)+P\left(C\right)-2P(B\cap C)-2P(C\cap A)-2P(A\cap B)+3P(A\cap B\cap C)$