180amid1solns - 180 Midterm 1 Solutions 1 Three dice are...

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180 Midterm 1 Solutions October 17, 2006 1) Three dice are rolled. Let A: all numbers are even B: all numbers are equal C: there is at least one 6 Compute P ( A B ) , P ( B C ) , P ( A B C ) , and P ( A B C c ) . Answer Note that P ( A B ) = P ( A ) + P ( B ) - P ( A B ) = 27 216 + 6 216 - 3 216 = 30 216 P ( B C ) = 1 216 P ( A B c ) = P ( A ) - P ( A B ) = 27 216 - 3 216 = 24 216 P ( A B C c ) = 2 216 To see this note that clearly there are 6 3 = 216 posibilities. Note that A B is the event that all three numbers are the same and even, and hence there are 3 such possibilities. B C is the event that all numbers are 6. Finally, A b C c means that not only are the numbers the same, and even, but they aren’t 6, so there are 2 remaining possibilities. 2) Consider an urn with 7 red balls and 3 blue balls. 1. Suppose we draw 4 balls without replacement and let X 1 be the total number of red balls we get. Compute P ( X 1 1) . 2. Suppose we draw 4 balls with replacement and let X 2 be the total number of red balls we get. Compute P ( X 2 2) . 3. Suppose we draw balls with replacement until we get a red ball and let X 3 be the total
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