Theorems of Probability

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Theorem 1:(Addition Rule of Probability)

If A and B are any two events, then

Proof:

(From the Venn diagram)

( A and A^C 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(A^C) = 1 - P(A)

Proof:

\ P(A) = 1 - P(A^C)

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}

E^C= {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