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PROBABILITY: Combining Two Events Jonathan Fei

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Definitions RANDOM EXPERIMENT: any procedure or situation that produces a definite outcome that may not be predictable in advance OUTCOME: a single possible result from a random experiment EVENT: Any collection of outcomes Siegel, Andrew F. and Charles J Morgan, Statistics and Data Analysis: An introduction John Wiley & Sons, Inc. 1996

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Example RANDOM EXPERIMENTS Flipping a coin 5 times and counting the number of heads OUTCOME You get 2 Heads and 3 Tails EVENT Getting more Heads than Tails

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OR When one, the other, or both events are true If Dave has a red shirt OR a green hat, then: He has a red shirt He has a green hat He has a red shirt and a green hat

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Mutually Exclusive When two events cannot occur at the same time Same as disjoint When you flip a coin, getting heads and getting tails are mutually exclusive. You either get heads or get tails.

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AND When two events are both true If Dave has a red shirt AND a green hat, then: He has a red shirt and a green hat

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Formulas P(A or B) = P(A) + P(B) – P(A and B) P(A and B) = P(A) + P(B) – P(A or B) If mutually exclusive, P(A or B) = P(A) + P(B)

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Conditional Probabilities For P(A given B) or P(A | B): It is the probability of A if you know the probability of B P(A | B) = P(A and B) / P(B) Example: In a standard 52 card deck, if you choose two cards and you know the first card was red, it changes the probability that the second card will be black.

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Independent Events Two events are independent if information about one does not affect the other If something is independent, then: P(A and B) = P(A) P(B) Example: If you flip a coin and get heads, it does not affect the chance of getting heads or tails on the next flip. The probability is the same.

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IMPORTANT DISTINCTIONS Mutually Exclusive and Independence are NOT the same thing Something mutually exclusive cannot be independent

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Example For example, the events A = being a senior in high school and B = being a freshman are mutually exclusive, since if I know that someone is a senior, they cannot be a freshman. In order for these two events to be independent, knowing that someone is a senior would have to not give me any information about whether a student is a freshman or not, which is obviously not the case. Todd Frost Flintridge Preparatory School

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