Probabilty

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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