ProbabilityProbability

Laws of Chance. Laws of Chance. Language of Uncertainty.Language of Uncertainty.

““The scientific interpretation of chance The scientific interpretation of chance

begins when we introduce probability.”begins when we introduce probability.”

-- David Ruelle-- David Ruelle

ProbabilityProbability

The notion of chance has existed for The notion of chance has existed for centuries.centuries. Egyptian tombs from around 2000 B.C.Egyptian tombs from around 2000 B.C. Card and Board games from 14Card and Board games from 14thth

centurycentury

Probability Quantifies Uncertainty.Probability Quantifies Uncertainty. 0 0 P(A) P(A) 1 1 Interpret P(A)=0 and P(A)=1Interpret P(A)=0 and P(A)=1

Basis of Inferential StatisticsBasis of Inferential Statistics

Classical Definition of ProbabilityClassical Definition of Probability

Let n be the total number of Let n be the total number of outcomes possible, and assume that outcomes possible, and assume that all outcomes are all outcomes are equally likelyequally likely..

Let m be the number of distinct Let m be the number of distinct outcomes that comprise the event A.outcomes that comprise the event A.

The probability of event A occurring The probability of event A occurring is:is:

P(A) = m / nP(A) = m / n

Theoretical ProbabilityTheoretical Probability

The classical definition of The classical definition of probability provides the theoretical probability provides the theoretical probability of event A. The probability of event A. The theoretical probability is not theoretical probability is not always calculable.always calculable. Examples:Examples:

In some situations, it is not possible to In some situations, it is not possible to count all outcomes.count all outcomes.

The outcomes are not equally likely to The outcomes are not equally likely to occur in all situations.occur in all situations.

Empirical ProbabilityEmpirical Probability

The empirical probability of an event is the The empirical probability of an event is the observed relative frequency of occurrence of observed relative frequency of occurrence of that event if the experiment is repeated that event if the experiment is repeated many times.many times.

The empirical probability converges to the The empirical probability converges to the theoretical probability (truth) as the number theoretical probability (truth) as the number of repetitions gets large.of repetitions gets large.

Experiment of Reps of #

A of sOccurrence of #)( AP

Probability TerminologyProbability Terminology ExperimentExperiment

an activity resulting in an uncertain outcomean activity resulting in an uncertain outcome Sample Space (S)Sample Space (S)

set of all possible outcomes in an set of all possible outcomes in an experimentexperiment

Event (A)Event (A) set of some of the possible outcomes of an set of some of the possible outcomes of an

experimentexperiment Any event is a subset of the sample spaceAny event is a subset of the sample space An event is said to occur if the outcome of the An event is said to occur if the outcome of the

experiment is a member of it.experiment is a member of it.

Probability NotationProbability Notation P(A) – denotes the probability of event P(A) – denotes the probability of event

A occurring ( 0 A occurring ( 0 P(A) P(A) 1 ) 1 ) n(A) – denotes the number of distinct n(A) – denotes the number of distinct

outcomes in event Aoutcomes in event A

Classical Definition of ProbabilityClassical Definition of Probability::

)(

)()(

Sn

AnAP

Complement of an EventComplement of an Event The complement of event A (denoted The complement of event A (denoted

A’) contains all elements in the A’) contains all elements in the sample space that are not in A.sample space that are not in A. A’ occurs when A does not occur.A’ occurs when A does not occur.

Complement Rule:Complement Rule:

Many problems are easier to solve using Many problems are easier to solve using the complement.the complement.

)'(1)( APAP

Discrete Probability Discrete Probability DistributionsDistributions

A discrete probability distribution A discrete probability distribution specifies the probability associated specifies the probability associated with each possible distinct value of with each possible distinct value of the random variable.the random variable.

A probability distribution can be A probability distribution can be expressed in the form of a graph, expressed in the form of a graph, table or formula.table or formula. For example: Let X be the number of For example: Let X be the number of

heads that you get when you flip 2 fair heads that you get when you flip 2 fair coins.coins.

Probability FunctionProbability Function A probability function, denoted P(x), A probability function, denoted P(x),

assigns probability to each outcome assigns probability to each outcome of a discrete random variable X.of a discrete random variable X.

Properties:Properties:

1 .21)(0 .1

P(x)xP

Binomial Probability DistributionBinomial Probability Distribution Results from an experiment in which a Results from an experiment in which a

trial with two possible outcomes is trial with two possible outcomes is repeated n times.repeated n times.

Heads/Tails, Yes/No, For/Against, Cure/No Heads/Tails, Yes/No, For/Against, Cure/No CureCure

One outcome is arbitrarily labeled a One outcome is arbitrarily labeled a success and the other a failuresuccess and the other a failure

Assumptions:Assumptions:1.1. n independent trialsn independent trials

2.2. Probability of success is p in each trialProbability of success is p in each trial

(so q=1-p is the probability of failure)(so q=1-p is the probability of failure)

Binomial Random VariableBinomial Random Variable Let X be the number of success in n Let X be the number of success in n

trials, then X is a binomial random trials, then X is a binomial random variable.variable.

Often, p is defined to be the proportion Often, p is defined to be the proportion of the population with a characteristic of of the population with a characteristic of interest, and X is the number sampled interest, and X is the number sampled with that characteristic of interest.with that characteristic of interest.

Probability FunctionProbability Functionxnx

xn qpCxP )(

P(x) = • px • qn-xn ! (n - x )! x!

Number of outcomes with

exactly x successes

among n trials

Probability of x successes

among n trials for any one

particular order

Binomial Probability FormulaBinomial Probability Formula

Binomial Mean, Var. & St. Dev.Binomial Mean, Var. & St. Dev. The mean, variance and standard The mean, variance and standard

deviation of a binomial random deviation of a binomial random variable with n trials and probability variable with n trials and probability of success p:of success p:

2

npnpq

npq