Probability

Definitions

• Probability – the likelihood of the occurrence of an event

• Random outcome – an outcome that cannot be predicted with certainty

• Statistical experiment or observation – any activity that results in a definite, but random, outcome

• Simple event – the most basic outcomes in a sample space; cannot be broken down any further

• Event – any simple event or collection of simple events in a sample space

• Sample space – the collection of all simple events for a statistical experiment

Classical Probability

▪ The sample space is a collection of equally likely outcomes

[pic]

Empirical Probability (relative frequency)

▪ The outcomes of a random experiment are observed over repeated trials

[pic]

Note: the Law of Large Numbers states that relative frequency gets closer and closer to the true probability as the sample size increases

Subjective Probability

▪ The probability of any event is a person’s opinion (educated guess? intuition?) of the likelihood of an event

P(A) = whatever you think it is!

Probability Rules!

1. The probability of any event must be between 0 and 1. That is, [pic] for any event A.

2. The sum of the probabilities for all simple events in a sample space must be 1.

3. The complement of event A consists of all outcomes in the sample space that do not make up event A, therefore

[pic][pic]

Note: A Venn diagram is useful for displaying relationships among events in a sample space.

The Venn diagram to show A and AC might look like this:

A AC

S

4. Two events are mutually exclusive (or disjoint) if they contain no common outcomes.

The Venn diagram to show two disjoint events A and B might look like this:

A B

S

5. The union of two events A and B consists of all outcomes in the sample space...