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Basic Probability

Statistics 515 Lecture 04

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Importance of Probability

• Modeling randomness and measuring uncertainty

• Describing the distributions of populations

• Obtaining descriptive measures of populations

• Assessing uncertainty in the sampling process

• Inference from sample to population

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Basic Elements• (Random) Experiments: “any activity!”

• Sample Space and Sample Points– Venn Diagrams– Tree diagrams– discrete versus continuous

• Events: Simple and Compound

• Probability of Events and Interpretation

• Two obvious properties of probability

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Some Examples of Experiments

• Toss a coin 3 times.

• A box has 100 balls, 1 red and 99 blues. Keep drawing a ball, with replacement, until the red ball is obtained.

• Pick 2 people from the group: {Pete, Pauly, Mary, Magdy}

• Super Bowl!– Who wins?

– Margin of Victory?

– Length of game in minutes.

– Yardage for M. Faulk.

• Credit card balance of a randomly chosen USC student.

• Genotype of progeny from heterozygote parents.

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More Examples

• Earthquakes– how many occurs in one year in the US of magnitude at least 2.0?

– Intensity of the next earthquake in CA

– Length of time between occurrences

• Genotype of progeny from heterozygote parents

• % return on an investment after one year

• How long will the “Weakest Link” show last?

• Many more!

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Interpretation of Probability

• Frequentist interpretation– “long-run” relative frequency– weak law of large numbers

• Theoretical interpretation– derived from a priori considerations

• Subjective interpretation– ‘personal’ beliefs

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Assigning Probabilities

For “equally likely” discrete sample spaces, which are sample spaces in which the sample points all have the same probability or chance of occurring,

)(

)()(

SN

ANAP

where N(A) = number of sample points in event A, and N(S) = number of sample points in the sample space.

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Some Illustrations

• A fair die is to be rolled once. What are the probabilities of each of the sample points?

• Suppose now however that two fair dice are to be rolled once simultaneously. What will be the sample space, and what are the appropriate probabilities of each of the sample points of the sample space?

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Another Example

• Let us classify the students in this class by Gender [Male (M) or Female (F)] and by Class [Undergraduate (U) or Graduate (G)]. Suppose that the numbers per Gender-Class combinations are as follows:

• N(M,U) = 16, N(M,G) = 4, N(F,U) = 22, N(F,G) = 7.

• Experiment: Draw one student at random.

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Some Questions

• What is the probability that a (male, graduate) student is chosen?

• What is the probability that a male student is chosen?

• What is the probability that the student is either a male or a graduate student?

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“Non-Equally” Likely Discrete Sample Spaces

To assign probabilities to the sample points of such sample spaces, the three conditions that should be satisfied by the probabilities assigned are:• each probability must be at least zero;• the sum of all the probabilities must equal 1;• the probability assignment should reflect a priori or subjective considerations, or long-run relative frequency expectations.

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An Illustration

• A biased die is to be rolled once. The die is biased in such a way that the probability of a face is proportional to the number of dots of that face. What are the appropriate probabilities of each of the faces?

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Another Example

• A box has 100 balls, with 1 ball red, and 99 balls blue (“a lottery model”). Suppose the experiment is to keep drawing a ball from this box, with replacement, until a red ball is obtained.

• What is the sample space and how do we assign probabilities to the sample points?

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Probability Operations

• Complements and Complementation Rule

• Union of Events and the Addition Rule for Mutually Exclusive Events

• Intersection of Events

• Addition Rule for Non-Mutually Exclusive Events

• DeMorgan’s Rules

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Conditional Probability

• Idea behind the notion of conditional probability

• Utility: probability updating

• Definition: P(B|A) = P(AB)/P(A)

• Reason for this definition

• Some Examples and Applications

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Multiplication Rule

• P(AB) = P(A)P(B|A)

• P(ABC) = P(A)P(B|A)P(C|AB)

• Applications of this concept

• Revisiting the “lottery model”

• Independent Events: P(AB) = P(A)P(B)

• When do independent events arise?

• Computing probabilities with independence

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Updating Probability

• Description of Setting

• Concrete Situation

• Theorem of Total Probabilities

• Tree Diagram Demonstration

• Bayes Theorem

• Prior and Posterior Probabilities

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An Example• There are two coins in a box, with one coin being

an ordinary coin (has a Head and a Tail), while the other coin being a two-headed coin. We draw one of these coins at random, and without looking which type of coin it is, we toss it once.

• What is the probability of getting a “Head” in this toss?

• Suppose that the outcome of the toss is a “Head.” What is the probability that we actually tossed the two-headed coin?

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Application On Medical Testing

• A test for HIV-infection has the following characteristic: If the person is infected, the test will have a positive result with probability of 0.9999; while if the person is not infected, the test will return a positive outcome with probability of 0.001. In a certain population, 5 in every 10000 are HIV-infected. Suppose that one person from this population is chosen at random and subjected to this test. If the test turned out to be positive, what is the probability that this person is HIV-infected?

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First, A Test of Our Intuition!

• Without yet computing the probability, who thinks that the probability that this person is HIV-infected is above .90?

• Who thinks that the probability that this person is HIV-infected is between 0.40 and 0.90?

• Who thinks that the probability is less than 0.40?