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Probability

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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  1. Probability • Laws of Chance. • Language of Uncertainty. “The scientific interpretation of chance begins when we introduce probability.” -- David Ruelle

  2. 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

  3. 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

  4. 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.

  5. 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.

  6. 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.

  7. 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:

  8. 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.

  9. 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.

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

  11. 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)

  12. 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

  13. 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

  14. Binomial Mean, Var. & St. Dev. • The mean, variance and standard deviation of a binomial random variable with n trials and probability of success p:

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