Probability and Probability Distributions

PROBABILITY  AND  PROBABILITY  DISTRIBUTIONS  –  REVIEW

Topics Outline

∙ Probability of Events                    

∙ Probability Rules

∙ Random Variables

∙ Probability Distributions

Probability of Events

There are many interpretations of probability. The three most widely used approaches are:

1. Classical method – based on the assumption of equally likely events.

                    Example: Six-sided fair die.

                    Each side has the same chance of turning up. Therefore, each has a probability 1/6.

2. Empirical method – based on experimental or historic data.

                    Example: Predicting the weather.

                    A 30% chance of rain today means that it rained on 30% of all days with similar

                    atmospheric conditions.

3. Subjective method – based on judgment, experience, intuition.

                    Example: Chris and Sally make an offer to purchase a house.

                    Sally believes that the probability their offer will be accepted is 0.8.

                    Chris, however, believes that the probability their offer will be accepted is 0.6.

In the case of equally likely events, a convenient way of thinking about probabilities is:

P(A) = [pic 1]

Probability Rules

No matter which method is used to assign probabilities to events, the following

probability rules (axioms) must hold.

Rule 1. [pic 2] for any event A

             That is, the probability of any event A is a number that lies between 0 and 1.

Rule 2. P(all outcomes) = 1

            That is, the probability of “something happening” is 1.

The compliment Ac of A is the event consisting of all sample points that are not in A.

That is, Ac is the event that A does not occur.

Rule 3. Compliment rule

P(Ac) = 1 – P(A)

That is, the probability of an event not occurring is 1 minus the probability that the event does occur.

Rule 4. Addition rule

P(AB) = P(A) + P(B) – P(AB)[pic 3][pic 4]

AB  = union          =   “or”  = either A or B or both[pic 5]

AB  = intersection = “and” = both A and B  [pic 6]

Two events A and B are mutually exclusive (disjoint) if they have no outcomes in common and so can never happen together.

Addition rule for mutually exclusive events

If A an B are mutually exclusive,

 P(AB) = P(A) + P(B)[pic 7]

Addition rule for more than two mutually exclusive events:

P(A1A2 . . . An) = P(A1) + P(A2) + . . . + P(An)[pic 8][pic 9][pic 10]

The conditional probability of the event A given that the event B has already occurred is given by

[pic 11]

Rule 5. Multiplication rule

[pic 12]

Two events A and B are independent if knowledge of the occurrence of one has no influence on the probability of occurrence of the other, that is

P(A|B) = P(A)

Multiplication rule for independent events

P(AB) = P(A)P(B)[pic 13]

Mutually exclusive events are not independent! If two events are mutually exclusive,

then knowledge of the occurrence of one has influence on the probability of the other.

(If you know that event B has occurred, you know that event A cannot have occurred).

How to check independence?

Use any one of the following techniques:

1. Check if A and B are mutually exclusive.

    If they are mutually exclusive, then they are not independent.

2. Check if the multiplication rule for independent events P(A B) = P(A)P(B) holds.[pic 14]