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handout_4_5_fall11(2) - 4.5 General Probability Rules...

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4.5 General Probability Rules Previous Probability Rules 1) The probability P( A ) of any event A satisfies 0 P( A ) 1. 2) All possible outcomes together must have probability 1. P( S ) = 1. 3) Two events A and B are disjoint →P( A or B ) = P( A ) + P( B ) 4) The complement rule states that→ P( A c ) = 1 – P( A ) 5) Two events A and B are independent P( A and B ) = P( A )P( B ) General addition rules The union of any collection of events is the event that at least one of the collection occurs. The intersection of any collection of events is the event that all of the events occur. Addition rule for disjoint events If events A , B , and C are disjoint in the sense that no two have any outcomes in common, then P (one or more of A , B , C ) = P( A ) + P( B ) + P( C ) P (union of A , B , C ) = P( A ) + P( B ) + P( C ) This rule extends to any number of disjoint events.
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General addition rule for unions of two events For any two events A and B , P( A or B ) = P( A ) + P( B ) – P( A and B ) Example #1 Consider tossing a die once. Let A be the event that you get a 1. Let B be the event that you get an odd number. What is the P( A or B )? Here, A and B are not disjoint. ●before (restrictive rule (4.2), holds only if A and B are disjoint): P( A or B ) = P( A ) + P( B ) ●now (general rule (4.5), holds whether or not A and B are disjoint): P( A or B ) = P( A ) + P( B ) – P( A and B ) Example #2 Consider tossing a die once. Let A be the event that you get a 1. Let B be the event that you get a 2.
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