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?

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

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

I think you are right. www.ddhw.com

sean9991

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