Chapter 5

Experiment

A process that leads to the occurrence of one and only one of several possible results. (Roll a die)

Bank of America customers can create their own 4 digit pin numbers, what is the likelihood that a customer will pick 2367?

(1/10)^4

Inferential statistics examples

- Toys and Things, a toy and puzzle manufacturer, recently developed a new games based on sports trivia. It wants to know whether sports buffs will purchase the game. "Slam Dunk" and "Home Run" are two of the names under consideration. To investigate, the president of Toys and Things decided to hire a market research firm. The firm selected a sample of 800 consumers from the population and asked each respondent for a reaction to the new game and its proposed titles. Using the sample results, the company can estimate the proportion of the population that will purchase the game.

If the set of events is collectively exhaustive and the events are mutually exclusive, the sum of the probabilities is:

1

Probability can not be greater than:

1

Some people are in favor of reducing federal taxes to increase consumer spending, and others are against it. Two persons are selected and their opinions are recorded. Assuming no one is undecided, list the possible outcomes.

1. A A 2. F F 3. A F 4. F A

Approaches to Assigning Probabilities

1.) Classical, 2.) Empirical, 3.) Subjective

Event

A collection of one or more outcomes of an experiment (i.e. observe an even number when rolling die.)

A Contingency Table is:

A cross-tabulation that simultaneously summarizes two variables of interest and their relationship.

Combination

A formula to count the number of possible arrangements when the order of the outcomes is not important. {c,b,a} is considered the same as {a,b,c}

Outcome

A particular result of an experiment (Observe a 1 when rolling die.)

Joint Probability

A probability that measures the likelihood two or more events will happen concurrently.

Special Rule of Addition

A rule used to find the probabilities of events made up of A or B when the events are mutually exclusive. (When one event occurs the other event can not.)

Contingency Table

A table used to classify sample observations according to two or more identifiable characteristics.

Probability

A value between zero and one, inclusive, describing the relative possibility (chance or likelihood) an event will occur.

Tree Diagram

A visual that is helpful in organizing and calculating probabilities for problems.

A sample of 40 oil industry executives was selected to test a questionnaire. One question about environmental issues required a yes or no answer. -A) What is the experiment? -B) List one possible event -C) Ten of the 40 executives responded yes, what is the probability that the next executive will respond yes? -D) What concept of probability does this represent? -E) Are each of the outcomes equally likely and mutually exclusive?

A) The 40 executives B) 26 or more people respond no, for example. C) 10/40=.25 D) Empirical E) The events are not equally likely but they are mutually exclusive.

Probability theory

Allows the decision maker to analyze the risks and minimize the gamble inherent, for example, (in marketing a new product or accepting an incoming shipment possibly containing defective parts.)

Permutation

Any arrangement of r objects selected from a single group of n possible objects.

Collectively Exhuastive

At least one of the events must occur when an experiment is conducted. (If an experiment has a set of events that includes every possible outcome, such as the events "an even number" and "an odd number" in the die-tossing experiment, every outcome will be either even or odd. So the set is collectively exhuastive.)

Classical Probability

Based on the assumption that the outcomes of an experiment are equally likely. Probability of an event= number of favorable outcomes/ total number of possible outcomes. "an even number of sports appear face up when rolling die" three "favorable" outcomes (a two, a four, and a six) therefor 3/6 = .5

Empirical Probability Formula

Empirical Probability= Number of times the event occurs/ Total number of observations.

A probability is frequently expressed as a:

Fraction 7/10, decimal .70, or a percentage 70%

Multiplication Forumula

If there are m ways of doing things and n ways of doing another thing, there are mxn ways of doing things. Total number of arrangements= (m)(n)(o)

Special rule of addition used for:

Mutually exclusive. Joint probability is P(A or B) is 0.

General rule of addition used for:

Not mutually exclusive

Law of Large Numbers

Over a large number of trials, the empirical probability of an event will approach its true probability.

General Rule of Multiplication

P(A and B) = P(A)P(B|A)

Special Rule of Multiplication Formula

P(A and B)= P(A)P(B)

General Rule of Addition

P(A or B) = P(A) + P(B) - P(A and B)

Special Rule of Addition formula

P(A or B) = P(A)+P(B) -If given three mutually exclusive events: P(A or B or C)= P(A)+P(B)+P(C)

Complement Rule formula

P(A)= 1- P(~A)

If events A and B are mutually exclusive and collectively exhaustive, which one of the following is true?

Probability of A= 1 - Probability of B

Two Rules of multiplication

Special Rule General Rule

Subjective Concept of Probability

The likelihood (probability) of a particular event happening that is assigned by an individual based on whatever information is available. 1.) Estimating the likelihood that the New England Patriots will play in the Super Bowl next year. 2.) Estimating the likelihood you will be involved in an automobile accident during the next 12 months. 3.) Estimating the likelihood the U.S. budget deficit will be reduced by half in the next 10 years.

Inferential statistics

The methods used to estimate a property of a population on the bases of a sample.

Independence

The occurrence of one event has no effect on the probability of the occurrence of another event. (If event A occurs, does A have any effect on the likelihood that event B occurs? If no, then A and B are independent.) ((The outcome of a coin toss (head or tails) is unaffected by the outcome of any other prior coin toss (head or tail).

Mutually Exclusive

The occurrence of one event means that none of the other events can occur at the same time. (A manufactured part is acceptable or unacceptable at the same time. In a sample of manufactured parts, the event of selecting an unacceptable part and the event of selecting an acceptable part are mutually exclusive.)

Conditional Probability

The probability of a particular event occurring, given that another event has occurred.

Empirical Probability

The probability of an event happening is the fraction of the time similar events happened in the past.

Odds

This means that in a total of seven trials (5+2), the event will occur five times and not occur two times. 5/(5+2) or 5/7 If the odds in favor of an event are (x) to (y), the probability of the event is x/(x+y)

Special Rule of Multiplication requires:

Two events A and B are independent. Two events are independent if the occurrence of one event does not alter the probability of the occurrence of the other event.

Rules of Multiplication

Used for finding the likelihood of two events happening.

Combination Formula

nCr = n!/r!(n-r)!

Permutation Formula

nPr = n!/(n-r)!

Complement Rule

the probability of an event occurring is 1 minus the probability that it doesn't occur