Chapter 5 Statistics Notes

General Multiplication Rule

P(E and F) = P(E)*P(F/E)

two events are ___ if the occurrences of event E in a probability experiment affects the probability of event F

dependent

An______ is an event that has a low probability of occurring

unusual event

What does the Law of Large Numbers tell us?

As the number of repetitions of a probability experiment increases, the proportions with which a certain outcome is observed gets closer to the probability of that outcome.

What is the definition of experiment?

Experiment is any process with uncertain results that can be repeated.

The General Addition Rule

For any two events E and F, P(E or F)= P(E) + P(F)-P(E and F)

Addition Rule for Disjoint Events

If E and F are disjoint events, then P(E or F) = P(E) + P(F)

Compliment Rule

If E represents any event and E^C represents the complement of E, then P(E^C)=1-P(E)

Computing Probability using the Classical Method

P(E) = (number of ways that E can occur)/ (number of possible outcomes) or m/n

Permutation with non distinct items

The number of permutations of n objects in which n1 are of one kind, n2 are of the second kind, and nk is of the kth kind is given by... n! divided by n1!*n2!*nk!

______ diagrams are used to represent events as circles enclosed in triangles

Venn

two events are _ if they have no outcomes in common.

disjoint

The long-term proportion in which a certain outcome is observed is________

the probability of that outcome

Disjoint events versus independent events

two events are disjoint if they have nothing in common while two events are independent if one event occurring does not promote the probability of the other event occurring

If an event is impossible, the probability of an event equals_

zero

What is the definition of an event

An event is any collection of outcomes from a probability experiment. An event consists of one outcome or more than one outcome. We will denote events with one outcome, sometimes called simple events, ei. In general, events are denoted using capital letters such as E.

First two rules of probabilities

1. The probability of any event, E, P(E), must be greater than or equal to zero and less than or equal to one. 2. The sum of the probabilities of all outcomes equals one. That is, if the sample space equals e1, e2, ......en then P(e1)+P(e2)+.......P(En)=1

Knowledge to recall

If small random samples are taken from large populations without replacement, it is reasonable to assume independence of the events. As a rule of thumb, if the sample size is less than five percent the population size, we treat the events as independent.

Compliment of an Event

Let S denote the sample space of a probability experiment and let E denote an event. The compliment of E, denoted E^C is all outcomes in a sample space S that are not outcomes of event E

Multiplication Rule for Independent Events

P(E and F)=P(E)*P(F)

Approximating Probabilities Using the Empirical Approach

P(E) ~relative frequency of E~ (frequency of E)/(number of trials of experiment)

Two events E and F are independent if

P(E/F)=P(E) and P(F/E)=P(F)

Multiplication Rule for n independent events

P(E1 and E2 and E3) = P(E1) * P(E2) *P(E3)

Conditional Probability Rule

P(F/E)=P(E and F)/P(E0

Probability Model

Probability Model lists all the possible outcomes of a probability experiment and each outcome's probability. A probability model must satisfy rule 1 and 2 of the rules of probabilities.

What is the definition of probability?

Probability deals with experiments that yield random short term results or outcomes yet reveal long term predictability.

What is the definition of sample space?

Sample Space, S of a probability experiment is the collection of all possible outcomes.

Conditional Probability

The notation P(F/E) is read the probability of event F given event E. It is the probability that the event F occurs given that the event E has occurred.

permutation

a permutation is an ordered arrangement in which r objects are chosen from n distinct objects and repetition is not allowed The symbol nPr represents the number of permutations of r objects selected from n objects.

Contigency table

also called two-way table because it relates to two categories of data

To be an unusual event it must have a probability less than _%

five

Two events E and F are ___ if the occurrences of event E in a probability does not affect the probability of event F

independent

combination

is a collection without regard to order in which r objects are chosen from n distinct objects and without repetition. The symbol nCr represents the number of combinations of n distinct objects taken r at a time.

Another name for disjoint events is ____ ______events

mutually exclusive

Example of mutually exclusive events

odds and evens

subjunctive probability

of an outcome is a probability obtained on the basis of personal judgement

If an event is a certainty the probability of an event is_

one

What is the definition of outcome?

outcomes are random short term results

The____ and _____ of the problem also determine the probability that separates unusual events from not so unusual events.

researcher, context