C__work_280sol7

C__work_280sol7 - Solutions to Homework 7 April 7 2007 6.1...

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Solutions to Homework 7 April 7, 2007 6.1: 2 There are 5 choices here, so guessing right on one of them will increase the score. Hence, the probability of increasing the score from one guess is 1 5 . The probability of guessing a question wrong is 4 5 . The probability of guessing two questions wrong is 4 5 · 4 5 = 16 25 . So, the probability of increasing the score, i.e. not getting questions wrong, is 1 - 16 25 = 9 25 . 6.2: 5 (b) ( A B )) ( B C ) ( A C ). (Note that it is redundant to include A B C Do you see why?) (d) If at most one happens, then at least two do not happen. As in (b), you could say ( A B ) ( B C ) ( A C ) Another way is to say ( A - B - C ) ( B - A - C ) ( C - A - B ) A B C . There are many possible answers. For example, if S is defined as in the problem, then A B C could be replaced by S - ( A B C ). (e) The best way is to say ( A B C ) ( B A C ) ( C A B ). Is it clear why this is not the same as the answer to (d)? Comments: 1. Remember that events are themselves sets. That means that answers like { ( A, B ) , ( B, C ) , ( A, C ) } and { x | x ⊆ { A, B, C } and | x | ≤ 2 } , while close, are incorrect. It is a matter of distinguishing set from set of sets —two very different things. 2. You have to be careful with set notation. You can’t write things like A + B - 2 C . It makes sense to talk | A | + | B | - | A B | (as we do in the Inclusion-Exclusion Principle), since we can add and subtract cardinalities of sets, which are numbers. But we can’t add and subtract sets; they’re not numbers. 6.2: 9 Let A be any set. Then A ∪ ∅ = A and A and are disjoint (since A ∩ ∅ = ). We can invoke the third rule of the definition of probability measure, which says that Pr( A ∪ ∅ ) = Pr( A ) + Pr( ). But since A ∪ ∅ = A , we have Pr( A ) = Pr( A ∪ ∅ ). Thus, 1
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Pr( A ) = Pr( A ) + Pr( ), hence, Pr( ) = 0. Some people simply let A = S and used the reasoning above. That’s perfectly correct too. 6.2: 16 Note that the sets A - B and A B are disjoint; for any element that occurs in one set, it cannot occur in the other (by definition of set difference/intersection). Also note that ( A - B ) ( A B ) = A . Then by the definition of probability measure, we have Pr( A ) = Pr(( A - B ) ( A B )) = Pr( A - B ) + Pr( A B ). Rearranging this gives us the desired result. Note: It was necessary to say that A - B and A B are disjoint sets whose union is the set A . Points were deducted if you did not say so.
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