A Card Game

sean9991

You have 52 playing cards (26 red, 26 black).  You draw cards one by one.  A red card pays you $1.  A black one fines you $1.  You can stop any time you want.  Cards are not return to the deck after being drawn.  What is the optimal strategy in terms of maximizing expected payoff?

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sean9991

It's a good answer.  But can you figure out what should be the optimal strategy? www.ddhw.com

QL

I think you are right. www.ddhw.com

独木桥

Good try. But I think the equation shoud be e(r,b) = max{[e(r-1,b)+1]*r/(r+b) + [e(r,b-1)-1]*b/(r+b),0}www.ddhw.com     www.ddhw.com

QL

In general, let's assume there are r red cards and b black cards, and the expected earning using the optimal strategy is  e(r,b), then we have: e(r,b) = e(r-1,b)*r/(r+b) + e(r,b-1)*b/(r+b) with boundary condition: e(r,0) = r; e(0,b)=0 Then we can calculate e(r,b), and stop the game when e(r,b)<=0, where r and b are the number of cards remaining in the deck.  www.ddhw.com