Chapter 2

Combination formula

(n over k)= (P_k,n)/k!= n!/k!(n-k)!

Counting rules

-Ordered pairs: (O1,O2)

empirical evidence, based on the results of many such repeatable experiments

-indicates that any relative frequency of this sort will stabilize as the number of replications n increases. -That is, as n gets arbitrarily large, n(A)/n approaches a limiting value referred to as the limiting (or long-run) relative frequency of the event A.

Multiplication rule for P(A "and" B)

A and B are independent if and only iff P(A "and" B)=P(A)*P(B)

Event -simple -compound

Any collection (subset) of outcomes contained in the sample space S. An event is simple if it consists of exactly one outcome and compound if it consists of more than one outcome.

Bernoulli random variable

Any random variable whose only possible values are 0 and 1

Two types of random variables

Discrete and continuous

Independence of more than two events

Events A1,..,An are mutually independent if for every k (k=2,3,...,n) and every subset of indices i1,i2,...,ik The events are mutually independent if the probability of the intersection of any subset of the n events is equal to the product of the individual probabilities

Product rule for ordered pairs

If the first element or object of an ordered pair can be selected in n1 ways, and for each of these n1 ways the second element of the pair can be selected in n2 ways, then the number of pairs is n1n2.

Law of total probability

Let A1,...,Ak be mutually exclusive and exhaustive events. Then for any other event B, P(B)=P(B|A)P(A1)+...+P(B|A)P(Ak)= summation P(B|Ai)P(Ai)

Independence

Often the chance that A will occur or has occurred is not affected by knowledge that B has occurred, so that P(A|B)=P(A), they are dependent otherwise

Multiplication rule

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

For any two events A and B with P(B)>0, the conditional probability of A given that B has occurred is defined by

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

Permutations formula

P_k,n= n!/(n-k)!

Sample space abbreviation

S={N,D}, where N represents not defective, D represents defective, and the braces are used to enclose the elements of a set

Interpreting probabliity

The interpretation that is most frequently used and most easily understood is based on the notion of relative frequencies.

Conditional probability

The probabilities assigned to various events depend on what is known about the experimental situation when the assignment is made. 1. Make an initial probability assignment 2. Partial information relevant to the outcome of the experiment may become available 3. This information may cause us to revise some of our probability assignements For a particular event A, P(A) represent the probability assignment to event A --> P (A) is the original or unconditional probability of event A.

Union

Two events A and B, denoted by A U B or read as "A or B" is the event consisting of all outcomes that are either in A or in B or in both events.

E represents

a simple event

Random variables

a variable because different numerical values are possible and random because the observed value depends on which of the possible experimental outcomes results notation: (rv)

Permutation

an ordered subset

Discrete random variable

an rv whose possible values either constitute a finite set or else can be listed in an infinite sequence in which there is a first element, a second element, and so on ("countably" infinite)

Combination

an unordered subset

We want axiom list to be

as short as possible

axioms

basic properties

This relative frequency interpretation of probability is said to

be objective because it rests on a property of the experiment rather than on any particular individual concerned with the experiment.

Probability distribution or probability mass function (pmf) of a discrete rv is

defined for every number x by p(x)=P(X=x)=P p(x) must be greater than or equal to 0 and the sum of all possible x p(x)=1

Intersection

denoted by A "and" B, read as "A and B", is the event consisting of all outcomes that are in both A and B

The complement of an event A

denoted by A' is the set of all outcomes in S that are not contained in A

Sample space

denoted by S , is the set of all possible outcomes of that experiment.

Continuous random variable

if both the following apply 1. Its set of possible values consists either of all numbers in a single interval on the number line (possibly infinite in extent, e.g., [0,10] union [20,30]) 2. No possible value of the variable has positive probability, that is, P(X=c)= for any possible value c

Language of probability is

non repeatable

Let O/ denote the

null event (the event consisting of no outcomes whatsoever). When A and B= O/, A and B are said to be mutually exclusive or disjoint events

ratio n(A)/n is

relative frequency occurrence of the event A in the sequence of n replications

Squiggly S

sample space

P(A|B) represents

the conditional probability of A given that the event B has occurred. B is the "conditioning event"

Multiplication rule is useful when

the experiment consists of several stages in succession

n(A)

the number of replications on which A does occur

Objective of probability

the objective of probability is to assign to each event A a number P(A), called the probability of the event A, which will give a precise measure of the chance that A will occur.

Tree diagram

used in counting and probability problems and can be used to represent pictorially all the possibilities

X(s)=x meaning

x is the value associated with the outcome s by the rv X