Review the following concepts if necessary:

- Probability for Beginning Statistics
- Classical, Relative Frequency, and Subjective Probability Definitions for Beginning Statistics
- Marginal and Conditional Probabilities for Beginning Statistics
- Mutually Exclusive, Dependent, and Independent Events for Beginning Statistics
- Intersection Of Events and Union of Events for Beginning Statistics
- Bayes' Theorem for Beginning Statistics
- Permutations and Combinations for Beginning Statistics

### Experiment, Outcomes, and Sample Space

- An experiment consists of using a 25-question test instrument to classify an individual as having either a type A or a type B personality. Give the sample space for this experiment. Suppose two individuals are classified as to personality type. Give the sample space. Give the sample space for three individuals.
- At a roadblock, state troopers classify drivers as either driving while intoxicated, driving while impaired, or sober. Give the sample space for the classification of one driver. Give the sample space for two drivers. How many outcomes are possible for three drivers?

*Ans.* For one individual, S = {A, B}, where A means the individual has a type A personality, and B means the individual has a type B personality.

For two individuals, S = {AA, AB, BA, BB}, where AB, for example, is the outcome that the first individual has a type A personality and the second individual has a type B personality.

For three individuals, S = {AAA, AAB, ABA, ABB, BAA, BAB, BBA, BBB}, where ABA is the outcome that the first individual has a type A personality, the second has a type B personality, and the third has a type A.

*Ans.* Let A be the event that a driver is classified as driving while intoxicated, let B be the event that a driver is classified as driving while impaired, and let C be the event that a driver is classified as sober.

The sample space for one driver is S = {A, B, C}.

The sample space for two drivers is S = {AA, AB, AC, BA, BB, BC, CA, CB, CC}.

The sample space for three drivers has 27 possible outcomes.

### Tree Diagrams and The Counting Rule

- An experiment consists of inspecting four items selected from a production line and classifying each one as defective, D, or nondefective, N. How many branches would a tree diagram for this experiment have? Give the branches that have exactly one defective. Give the branches that have exactly one nondefective.
- An experiment consists of selecting one card from a standard deck, tossing a pair of dice, and then flipping a coin. How many outcomes are possible for this experiment?

*Ans.* The tree would have 2^{4} = 16 branches which would represent the possible outcomes for the experiment.

The branches that have exactly one defective are DNNN, NDNN, NNDN, and NNND.

The branches that have exactly one nondefective are NDDD, DNDD, DDND, and DDDN.

*Ans.* According to the counting rule, there are 52 × 36 × 2 = 3,744 possible outcomes.

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