Probability - I


   
 
Theorems of Probability
Theorem 1:(Addition Rule of Probability)
 
If A and B are any two events, then
 
 
Proof:
 
 
 
 
(From the Venn diagram)
 
 
( A and AC B are mutually exclusive)
 
 
 
 
 
 
 
Note 1:
 
 
If A and B are mutually exclusive events, then
 
P(A B) = P(A) + P(B)
 
Note 2:
 
If A, B, C are any three events, then
 
 
Example:
 
In tossing a fair die, what is the probability that the outcome is odd or grater than 4?
 
Suggested answer:
 
Let E1 be the event that the outcomes are odd.
 
E1 = {1,3,5}
 
Let E2 be the event that the outcomes are greater than 4.
 
E2 = {5,6}
 
 
 
 
 
 
 
Theorem 2:
 
P(AC) = 1 - P(A)
 
Proof:
 
 
 
 
\ P(A) = 1 - P(AC)
 
Example:
 
In tossing a die experiment, what is the probability of getting at least 2.
 
Suggested answer:
 
Let E be the event that the outcome is at least 2, then
 
E = {2,3,4,5,6}
 
EC= {1}
 
 
 
 
Theorem 3:
 
P(f) = 0
 
Proof:
 
The proof follows from theorem 2,
 
P(f)C = 1 - P(f)
 
 
 
= 1 - 1 = 0
 
 
Example:
 
In throwing a die experiment, what is the probability of occuring a number greater than 8 ?
 
Suggested answer:
 
Let E be the event where the outcome is greater than 8.
 
E = f
 
P(f) = 0
 
  
     
   
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