HW4Solutions (dragged) 4 - A = p A 1(1-p A(1-p B(1-p B p...

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Extra Problem 3 Let p 1 = P (A wins in A-B duel where A goes first) = P (A wins AB) and let A 1 = A kills B on first shot. Then p 1 = P (A wins AB | A 1) P ( A 1) + P (A wins AB | A 1 c ) P ( A 1 c ) = 1( p A ) + P (A wins AB | A 1 c )(1 - p A ) = p A + (1 - p A ) P (A wins AB | A 1 c ) and, letting B 1 = B kills A on first shot, P (A wins AB | A 1 c ) = P (A wins in A-B duel where B goes first) = P (A wins BA) = P (A wins BA | B 1) P ( B 1) + P (A wins BA | B 1 c ) P ( B 1 c ) = 0( p B ) + p 1 (1 - p B ) = p 1 (1 - p B ) Thus, p 1 = p A + (1 - p A ) p 1 (1 - p B ) ) p 1 = p A 1 - (1 - p A )(1 - p B ) and P (A wins AB | A 1 c ) = p A 1 - (1 - p A )(1 - p B ) (1 - p B ) and, since B will shoot C and C will shoot B, P (A wins | doesn’t hit C) = p B p 1 + p A (1 - p B ) Now, P (A wins) = P (A wins | hits C) P (hits C) + P (A wins | doesn’t hit C) P (doesn’t hit C) = P (A wins BA) p A + P (A wins | doesn’t hit C)(1
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Unformatted text preview: A ) = p A 1-(1-p A )(1-p B ) (1-p B ) p A + (1-p A )[ p B p A 1-(1-p A )(1-p B ) + p A (1-p B )] Extra Problem 4 The optimal strategy For the problem is a stopping rule. Under it, the player rejects the frst r-1 prizes (let prize M be the best prize among these r-1 prizes), and then selects the frst subsequent prize that is better than prize M. It can be shown that the optimal strategy lies in this class oF strategies. ±or an arbitrary cuto ↵ point r , the probability 5...
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