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