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Probability
by Chelsea Gray
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Probability

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Random Circumstances

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Random circumstance: one in which the outcome is unpredictable. In many cases the outcome is not determined until we observe it.

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Probability: a number between 0 and 1 that is assigned to a possible outcome of a random circumstance.

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Interpretations of Probability

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Relative frequency: In situations that we can imagine repeating many times, we define the probability of a specific out come as the proportion of times it would occur over the long run.

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Personal probability: the degree to which a given individual believes that the event will happen

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Subjective probability: The degree of belief may be different for each individual

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Coherent: your personal probability of one event doesn't contradict your personal probability of another

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Probability Definitions and Relationships

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Sample space: the collection of unique, non-overlapping possible outcomes of a random circumstance.

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Complementary event: the two events do not contain any of the same simple events and together they cover the entire sample space.

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Mutually exclusive/disjoint: the events do not contain any of the same simple events (outcomes).

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Independent event: knowing that one will occur (or has occurred) does not change the probability that the other occurs.

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Dependent event: knowing that one will occur (or has occurred) changes the probability that the other occurs.

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Conditional probability: of the event B, given that the event A occurs, is the long-run relative frequency with which event B occurs when circumstances are such that A also occurs.

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Basic Rules for Finding Probabilities

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Rule 1 ( "not the event"): to find the probability of A^c, the complement of A, use P(A^c) = 1 - P(A). P(A^c) is the probability that event A won't occur

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Rule 2 (addition rule for "either/or"): to find the probability that either A or B or both happen.

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Rule 3 (multiplication rule for "and"): to find the probability that two events A and B both occur simultaneously or in a sequence.

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Rule 4 (conditional probability): an algebraic restatement of rule 3a

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Sample with replacement: individuals are returned to the eligible pool for each selection.

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Sample without replacement: individuals are not eligible for subsequent selection.

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Strategies for Finding Complicated Probabilities

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Bayes Rule: P(A|B) = P(A and B)/P(B|A)P(A) + P(B|A^c)P(A^c)

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Tree diagram: a schematic representation of the sequence of events and their probabilities, including conditional probabilities based on previous events that happen sequentially.

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