Chapter 5

n!

# ways of n different identifiable items can be arranged on a line

Counting Rules- Combination n: r: Ex:

-- An arrangement where the order of the objects selected from a specific pool of objects is not important. n: total # objects r: total # objects selected (same as permutation) Ex: The outcome (a,b,c) is considered to be same as (b,a,c)

Counting Rules- Permutation n: r: Example

--To find the number of possible arrangements when the order of outcomes is important. n: total # objects r: total # objects selected Ex: Three electronic parts, a transistor, an LED, and a synthesizer, are assembled into a plug in unit for a tv. The parts can be assembled in any order. How many different ways can the parts be associated?

Conditional Probability

Probability of a particular event occurring, given that another event has occurred. Probability of event B given that event A has occurred. P(B1A)

Marginal probability

Probability of one event occurring, P(b) p(a)

Joint Probability Example

Probability that measures likelihood two or more events will happen concurrently. Example: A tourist visiting both attraction being compared.

Experiment

Process that leads to occurrence of one and only one of several possible observations.

General Multiplication Rule

--Used to find the joint probability that two independent events will occur. --States that for two ind. events, A and B. The joint probability that both events happen is found by multiplying probability event A by conditional probability of event B occurring given A occurred.

Empirical probability and example

-Probability of event happening is fraction of the time similar events happened in the past. - Based on the law of large numbers - Larger # of observations provides a more accurate estimate of probability # times event occurs/ total observations Ex: Batting average

Tree Diagrams

-Useful for portraying conditional and joint probabilities. - Useful for analyzing business decisions involving several stages. -Graph helpful in organizing calculations involving several stages. Each segment of tree is one stage of problem. Branches are weighted by probabilities.

A space shuttle exploded for the second disaster in 123 missions. What is the probability that a future mission is successful.

121/123 = 0.98

5!, solve

5 (5-1) (5-2) (5-3) (5-4) = 120

Probability

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

Classical probability and equation example

Assumption that the outcomes of an experiment are equally likely. Equation: # favorable outcomes/ total #possible outcomes EX: the probability of rolling even number on die is 3/6.

Law of large numbers

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

The General Rule of Addition

If A and B are two events not mutually exclusive, then P (A or B) is given by formula: P(A or B)= P(A) + P(B) - P(A and B)

Collectively exhaustive events Example

If at least one of the events must occur when experiment is conducted. Ex: An even number and an odd number in a die toss

Independent events Example

If occurrence of one event does not affect occurrence of another event. Ex: The outcome of a coin toss is not affected by the previous tosses.

Mutually exclusive events Example

If the occurrence of any one event means none of the others can occur at the same time. Ex: The variable gender

Multiplication Counting Rule

If there are m ways to do one thing. And n ways of doing another thing. There are (M)(N) ways to do both.

Special rule of addition Formula

If two events A and B are mutually exclusive, the probability of one or the other events occurring equals the sum of their probabilities. P(A or B)= P(A) + P (B)

Subjective concept of probability What is it? When is it used? example

Likelihood of particular event happening that is assigned by an individual based on the info available. Used if there is little or no pat experience or info to base a probability. Ex: Estimating likelihood patriots will play in SuperBowl next year. Estimating likelihood you will be married before 30.

Special Rule of multiplication

Requires two events A and B are independent Two events A and B are independent if occurrence of one has no effect on occurrence of another. Formula: P(A and B) = P(A)P(B)

Outcome

Result of an experiment

Experiment example

Rolling die, counting number of board members over 60 years of age

Contingency Tables

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

Joint probability

The probability of events occurring together (or one after the other).

U meaning

Union, set out outcomes that are either in A or B or both.

The complement rule

Used to determine probability of event occurring by subtracting probability of event not occurring from 1. P(A)= 1 - P(~A)

When are two events independent

When the fractions are equal to one another

Three ways to assign probability

classical, empirical, subjective concept of probability

Event

collection of one or more outcomes of an experiment

events example

observe an even number, more than 13 board members are over 60

outcomes example

observing a 1, one board member is over 60

In an "and" statement what is multiplied by itself, the first or the second variable, what is the conditional

the first conditional: the second

go to other side

what is the symbol -- set of outcomes in A and B at the same time