hw3sol - Computer Science 340 Reasoning about Computation...

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Computer Science 340 Reasoning about Computation Homework 3 Due at the beginning of class on Wednesday, October 10, 2007 Problem 1 Suppose you toss three fair, mutually independent coins. Define the following events: A - the event that the first coin is heads B - the event that the second is heads C - the event that the third coin is heads D - the event that the number of heads is even Determine the maximum value of k such that these events are k -wise independent. Solution: These events are 3-wise independent. First, let us note that these events are not mutually independent and thus they are not 4-wise independent. Indeed, Pr( A B C D ) = 0 . Here, we used the fact that if the events A , B and C happen, then the number of heads is odd, so the event D does not happen. On the other hand Pr( A ) · Pr( B ) · Pr( C ) · Pr( D ) = 1 16 , thus Pr( A B C D ) = Pr( A ) · Pr( B ) · Pr( C ) · Pr( D ) . Now we prove that any three of the events A , B , C and D are mutually independent. The events A , B , C are mutually independent by the condition. Let us prove that A , B , and D are mutually independent. Let I A , I B , and I D be the indicators of the events A , B and D . Fix arbitrary x 1 , x 2 , x 3 ∈ { 0 , 1 } . Let us compute the following probabilities: Pr( I A = x 1 ) = 1 2 ; Pr( I B = x 2 ) = 1 2 ; Pr( I D = x 3 ) = 1 2 ;
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Pr( I A = x 1 & I B = x 2 & I D = x 3 ) = Pr( I A = x 1 & I B = x 2 & I A I B I C = x 3 ) = Pr( I A = x 1 & I B = x 2 & x 1 x 2 I C = x 3 ) = Pr( I A = x 1 & I B = x 2 & I C = x 1 x 2 x 3 ) = Pr( I A = x 1 ) · Pr( I B = x 2 ) · Pr( I
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