ISDS 2000 Chapter 4

probability

is a numerical value that measures the likelihood that an uncertain event occurs.

event

is a subset of the sample space.

experiment

is a trial that results in one of several uncertain outcomes.

Prior Probability

is the original (unconditional) probability (e.g., P(B) ). original / unconditional probabilty

Joint Probabilities

"Interior" table probabilities, (intersections)

Marginal Probabilities

"Marginal / Unconditional" probabilities, (row/column total)

definite

A probability of one indicates ____________ events.

impossible

A probability of zero indicates __________ events.

BAYES THEOREM

A procedure for updating probabilities based on new information. Given a set of prior probabilities for an event and some new information, the rule for updating the probability of the event is called prior and posterior probability

0.36 The addition rule is calculated as P(A∪B)=P(A)+P(B)−P(A∩B) . The multiplication rule is calculated as P(A∩B) = P(B|A) × P(A). Given: P(A) = 0.40 and P(B|A) = 0.60, P(B) = 0.20 P(A∪B) = 0.40 + 0.20 − 0.40 × 0.60 = 0.36.

Alison has all her money invested in two mutual funds, A and B. She knows that there is a 40% chance that fund A will rise in price, and a 60% chance that fund B will rise in price given that fund A rises in price. There is also a 20% chance that fund B will rise in price. What is the probability that at least one of the funds will rise in price?

Subjective probabilities

Draws on personal and subjective judgment. ex. At the beginning of the semester, John believes he has a 90% chance of receiving straight A's.

Dependent Events

Events are considered ________________ if the occurrence of one is related to the probability of the occurrence of the other.

False

Events are exhaustive if they do not share common outcomes of a sample space. True or False

Contingency Table

Frequencies for 2 QUALITATIVE variables with each cell representing mutually exclusive combinations of the pair.

Exhaustive

If all possible outcomes of a random experiment are included in the events.

Mutually exclusive

If they do not share any common outcome of a random experiment.

{TT, HH} and {TH, HT} Events are mutually exclusive if they do not share any common outcome of a random experiment. Events are collectively exhaustive if all possible outcomes of a random experiment are included in the events.

In an experiment in which a coin is tossed twice, which of the following represents mutually exclusive and collectively exhaustive events? Multiple Choice {TT, HH} and {TT, HT} {HT, TH} and {HH, TH} {TT, HH} and {TH, HT} {TT, HT} and {HT, TH}

The Total Probability Rule

P(A) is the sum of its intersections with some mutually exclusive and exhaustive events corresponding to an experiment. Consider event B and its complement Bc. These two events are mutually exclusive and exhaustive.

relative frequency

P(A)= the number of outcomes in A / the numbers of outcomes in S

Odds

Probabilities can be expressed as fractions, percentages, and ______.

The Combination Formula

The number of ways to choose x objects from a total of n objects, where the order in which the x objects are listed does not matter, nCx = (nx)= n! / (n-x)!x!

The Permutation Formula

The number of ways to choose x objects from a total of n objects, where the order in which the x objects is listed does matter, is nPx = n!/ (n-x)!

Conditional Probability

The probability of an event given that another event has already occurred.

Unconditional (Marginal) Probability

The probability of an event without any restriction.

Complement Rule

The probability of the complement of an event, P(Ac), is equal to one minus the probability of the event.

The Addition Rule

The probability that event A or B occurs, or that at least one of these events occurs, is

True

Two events A and B are independent if the probability of one does not influence the probability of the other. True or False

Independent Events

Two events are ______________ if the occurrence of one event does not affect the probability of the occurrence of the other event.

Empirical probability

a relative frequency of occurrence. This is an ______________ probability, since it is based on observed outcomes of an experiment. ex. A political reporter announces that there is a 40% chance that the next person to come out of the conference room will be a Republican, since there are 60 Republicans and 90 Democrats in the room.

sample space

denoted S, of an experiment includes all possible outcomes of the experiment. ex. containing letter grades is: s=(A,B,C,D,F)

Classical probability

logical analysis ex. Before flipping a fair coin, Sunil assesses that he has a 50% chance of obtaining tails.

complement

of event A (i.e., Ac) is the event consisting of all simple events in the sample space S that are not in A.

intersection

of two events (A ∩ B) consists of all simple events in both A and B.

union

of two events (A ∪ B) is the event consisting of all simple events in A or B.

Multiplication Rule

the probability that A and B both occur is equal to may also be used to determine independence. That is, two events are independent if the above equality holds.

posterior probability

with new information we update prior probability to include a condition.... conditional probability is the updated (conditional) probability (e.g., P(B | A) ).

combination formula

x objects n at a time / order DOES NOT matter.

permutation formula

x objects taken N at a time / order DOES matter