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25 Cards in this Set
- Front
- Back
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Statistical experiment
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An act or process of observation that leads to a single outcome that cannot be predicted with certainty.
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A sample point
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The most basic outcome of an experiment.
--AKA: simple event |
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Sample space
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The collection of all the sample points/simple events of an experiment.
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Probability of a sample point(informal definition)
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The probability of a sample point is a number between 0 and 1 which measures the likelihood that the outcome will occur when the experiment is performed.
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Probability Rules for Sample Points
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Let p(sub-i) represent the probability of sample point i. Then
1: All sample point probabilities must lie between 0 and 1 (i.e., 0 less than or equal to p(sub-i) less than or equal to 1). 2: The probabilities of all the sample points wihtin a sample sapce must sum to 1 (i.e., Sigma-p(sub-i) equals 1). *p(sub-i) means a small i by the bottom right corner of the p. |
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A probability "event"
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An event is a specific collection of sample points.
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Probability of an Event
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The probability of an event A is calculated by summing the probabilities of the sample points in the sample space for A.
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Steps for Calculating Probabilities of Events
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1: Define the experiment; that is, describe the process used to make an observation and the type of observation that will be recorded.
2: List the sample points. 3: Assign probabilities to the sample points. 4: Determine the collection of sample points contained in the event of interest. 5: Sum the sample point probabilities to get the probability of the event. |
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Combinatorial mathematics
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A branch of mathematics concerned with developing counting rules for given situations.
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Combinations Rule
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Suppose a sample of "n" elements is to be drawn from a set of "N" elements. Then the number of different samples possible is denoted by *(N over n): open paranthesis with a big N above a little n, close paranthesis* and is equal to (N over n) = N! divided by n!(N-n)! where n! = n(n-1)(n-2)...(3)(2)(1) and similarly for N! and (N-n)! For example, 5! = 5*4*3*2*1. [Note: The quanitity 0! is defined to be equal to 1.]
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Compound events
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A combination of two or more other events.
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Union compound events
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The union of two events "A" and "B" is the event that occurs if either "A" or "B" (or both) occurs on a single performance of the experiment. We denote the union of events "A" and "B" by the simple "A" U "B". "A" U "B" consists of all the sample points that belong to "A" or "B" OR both.
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Intersection compound events
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The intersection of two events A and B is the event that occurs if both A and B occus on a single performance of the experiment. We write "A (upside down U) B" for the intersection of A and B. "A (upside down U) B" consists of all the sample points belonging to both A and B.
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Complementary events
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The complement of an event A is the event that A does not occur--that is, the event consisting of all sample points that are not in event A. We donote the complement of A by A'.
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Rule of Complements
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The sum of the probabilities of complementary events equals 1: that is, P(A)+P(A^c)=1
*A^c means A with a small c in the upper right corner. |
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Additive Rule of Probability
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The probability of the union of events A and B is the sum of the probability of event A and the probability of event B, minus the probability of the intersection of events A and B; that is P(A U B)=P(A)+P(B)-P(A*upside down U*B)
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Mutually Exclusive events
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Events A and B are mutually exclusive if A*upside down U*B contains no sample points--that is, if A and B have no sample points in common. For mutually exclusive events, P(A*upside down U*B)=0
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Probability of Union of Two Mutually Exclusive Events
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If two events A and B are mutually exclusive, the probability of the union of A and B equals the sum of the probability of A and the probability of B; that is P(A U B)=P(A)+P(B)
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Unconditional Probabilities
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When no special conditions are assumed to affect the probability of outcomes other than those which define the experiment.
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Condition Probabilities
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When their is additional knowledge that might affect the outcome of an experiment, requiring an adjustment in measuring the probability of an event of interest.
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Conditional Probability Formula
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To find the conditional probability that event A occurs given that event B occures, divide the probability that both A and B occur by the probability that B occurs; that is, p(AlB)=P(A*upside down U*B) divided by P(B)
[We assume that P(B) does not equal 0] |
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Multiplicative Rule of Probability
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P(A *upside down U* B) = P(A)P(BlA) or, equivalently, P(A *upside down U* B) = P(B)P(AlB)
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Independent Events
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Events A and B are independent events if the occurrence of B does not alter the probability that A has occurred; that is, events A and B are independent if P(AlB) = P(A)
When events A and B are independent, it is also true that P(BlA) = P(B) Events that are not independent are said to be dependent. |
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Probability of Intersection of Two Independent Events
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If events A and B are independent, then the probability of the intersection of A and B equals the product of the probabilities of A and B; that is P(A *upside down U* B) = P(A)P(B)
The converse is also true: If P(A *upside down U* B) = P(A)P(B), then events A and B are independent. |
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Random Sample
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If "n" elements are selceted from a population in such a way that every set of "n" elements in the population has an equal probability of being selected, then the "n" elements are said to be a random sample.
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