Probability - University of Texas at Dallas

[Pages:25]Probability

OPRE 6301

Random Experiment. . .

Recall that our eventual goal in this course is to go from the random sample to the population. The theory that allows for this transition is the theory of probability.

A random experiment is an action or process that leads to one of many possible outcomes. Examples:

Experiment

Flip a coin Roll a die Exam Marks Course Grades Task completion times

Outcomes

Heads, Tails Numbers: 1, 2, 3, 4, 5, 6 Numbers: (0, 100) F, D, C, B, A Nonnegative values

The list of possible outcomes of a random experiment must be exhaustive and mutually exclusive.

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Sample Space. . .

The set of all possible outcomes of an experiment is called the sample space. We will denote the outcomes by O1, O2, . . . , and the sample space by S. Thus, in set-theory notation,

S = {O1, O2, . . .}

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Events. . .

An individual outcome in the sample space is called a simple event, while. . . An event is a collection or set of one or more simple events in a sample space. Example: Roll of a Die S = {1, 2, ? ? ? , 6} Simple Event: The outcome "3". Event: The outcome is an even number (one of 2, 4, 6) Event: The outcome is a low number (one of 1, 2, 3)

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Assigning Probabilities. . .

Requirements

Given a sample space S = {O1, O2, . . .}, the probabilities assigned to events must satisfy these requirements: 1. The probability of any event must be nonnegative,

e.g., P (Oi) 0 for each i. 2. The probability of the entire sample space must be 1,

i.e., P (S) = 1. 3. For two disjoint events A and B, the probability of

the union of A and B is equal to the sum of the probabilities of A and B, i.e.,

P (A B) = P (A) + P (B) .

Approaches

There are three ways to assign probabilities to events: classical approach, relative-frequency approach, subjective approach. Details. . .

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Classical Approach. . .

If an experiment has n simple outcomes, this method would assign a probability of 1/n to each outcome. In other words, each outcome is assumed to have an equal probability of occurrence. This method is also called the axiomatic approach. Example 1: Roll of a Die

S = {1, 2, ? ? ? , 6} Probabilities: Each simple event has a 1/6 chance of

occurring.

Example 2: Two Rolls of a Die

S = {(1, 1), (1, 2), ? ? ? , (6, 6)} Assumption: The two rolls are "independent." Probabilities: Each simple event has a (1/6) ? (1/6) =

1/36 chance of occurring.

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Relative-Frequency Approach. . .

Probabilities are assigned on the basis of experimentation or historical data.

Formally, Let A be an event of interest, and assume that you have performed the same experiment n times so that n is the number of times A could have occurred. Further, let nA be the number of times that A did occur. Now, consider the relative frequency nA/n. Then, in this method, we "attempt" to define P (A) as:

P

(A)

=

lim

n

nA n

.

The above can only be viewed as an attempt because it is not physically feasible to repeat an experiment an infinite number of times. Another important issue with this definition is that two sets of n experiments will typically result in two different ratios. However, we expect the discrepancy to converge to 0 for large n. Hence, for large n, the ratio nA/n may be taken as a reasonable approximation for P (A).

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Example 1: Roll of a Die

S = {1, 2, ? ? ? , 6}

Probabilities: Roll the given die 100 times (say) and suppose the number of times the outcome 1 is observed is 15. Thus, A = {1}, nA = 15, and n = 100. Therefore, we say that P (A) is approximately equal to 15/100 = 0.15.

Example 2: Computer Sales

A computer store tracks the daily sales of desktop computers in the past 30 days.

The resulting data is:

Desktops Sold

0 1 2 3 4 5 or more

No. of Days

1 2 10 12 5 0

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