# 01_21ans - STAT 400 Examples for Spring 2011 1 Suppose a...

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STAT 400 Examples for 01/21/2011 Spring 2011 1. Suppose a 6-sided die is rolled. The sample space, S , is { 1, 2, 3, 4, 5, 6 } . Consider the following events: A = { the outcome is even } , B = { the outcome is greater than 3 } , a) List outcomes in A, B, A ' , A B, A B. A = { the outcome is even } = { 2, 4, 6 } , B = { the outcome is greater than 3 } = { 4, 5, 6 } , A ' = { 1, 3, 5 } , A B = { 4, 6 } , A B = { 2, 4, 5, 6 } . b) Find the probabilities P( A ), P( B ), P( A ' ), P( A B ), P( A B ) if the die is balanced (fair). P( A ) = 3 / 6 , P( B ) = 3 / 6 , P( A ' ) = 3 / 6 , P( A B ) = 2 / 6 , P( A B ) = 4 / 6 .

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c) Suppose the die is loaded so that the probability of an outcome is proportional to the outcome, i.e. P( 1 ) = p , P( 2 ) = 2 p , P( 3 ) = 3 p , P( 4 ) = 4 p , P( 5 ) = 5 p , P( 6 ) = 6 p . i) Find the value of p that would make this a valid probability model. P(1) + P(2) + P(3) + P(4) + P(5) + P(6) = 1.0. p + 2 p + 3 p + 4 p + 5 p + 6 p = 21 p = 1.0. p = 1 / 21 . ii) Find the probabilities P( A ), P( B ), P( A ' ), P( A B ), P( A B ). P( A ) = P(2) + P(4) + P(6) = 2 / 21 + 4 / 21 + 6 / 21 = 12 / 21 . P( B ) = P(4) + P(5) + P(6) = 4 / 21 + 5 / 21 + 6 / 21 = 15 / 21 . P( A ' ) = 1 – P( A ) = 1 – 12 / 21 = 9 / 21 . OR P( A ' ) = P(1) + P(3) + P(5) = 1 / 21 + 3 / 21 + 5 / 21 = 9 / 21 .
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