# Probability - PowerPoint PPT Presentation

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Probability. Laws of Chance. Language of Uncertainty. “The scientific interpretation of chance begins when we introduce probability.” -- David Ruelle. Probability. The notion of chance has existed for centuries. Egyptian tombs from around 2000 B.C.

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Probability

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### Probability

• Laws of Chance.

• Language of Uncertainty.

“The scientific interpretation of chance

begins when we introduce probability.”

-- David Ruelle

### Probability

• The notion of chance has existed for centuries.

• Egyptian tombs from around 2000 B.C.

• Card and Board games from 14th century

• Probability Quantifies Uncertainty.

• 0  P(A)  1

• Interpret P(A)=0 and P(A)=1

• Basis of Inferential Statistics

### Classical Definition of Probability

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

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

• The probability of event A occurring is:

P(A) = m / n

### Theoretical Probability

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

• Examples:

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

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

### Empirical Probability

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

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

### Probability Terminology

• Experiment

• an activity resulting in an uncertain outcome

• Sample Space (S)

• set of all possible outcomes in an experiment

• Event (A)

• set of some of the possible outcomes of an experiment

• Any event is a subset of the sample space

• An event is said to occur if the outcome of the experiment is a member of it.

### Probability Notation

• P(A) – denotes the probability of event A occurring ( 0  P(A)  1 )

• n(A) – denotes the number of distinct outcomes in event A

• Classical Definition of Probability:

### Complement of an Event

• The complement of event A (denoted A’) contains all elements in the sample space that are not in A.

• A’ occurs when A does not occur.

• Complement Rule:

• Many problems are easier to solve using the complement.

### Discrete Probability Distributions

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

• A probability distribution can be expressed in the form of a graph, table or formula.

• For example: Let X be the number of heads that you get when you flip 2 fair coins.

### Probability Function

• A probability function, denoted P(x), assigns probability to each outcome of a discrete random variable X.

• Properties:

### Binomial Probability Distribution

• Results from an experiment in which a trial with two possible outcomes is repeated n times.

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

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

• Assumptions:

• n independent trials

• Probability of success is p in each trial

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

• ### Binomial Random Variable

• Let X be the number of success in n trials, then X is a binomial random variable.

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

• Probability Function

### Binomial Probability Formula

n!

P(x) = • px•qn-x

(n - x )! x!

Probability of x successes among n trials for any one particular order

Number of

outcomes with exactly x successes among n trials

### Binomial Mean, Var. & St. Dev.

• The mean, variance and standard deviation of a binomial random variable with n trials and probability of success p: