Theorem 1:(Addition Rule of Probability)
If A and B are any two events, thenProof:
(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, thenExample:
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, thenE = {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