1. copyright Robert J. Marks II ECE 5345 Multiple Random Variables: A Gambling Sequence

2. copyright Robert J. Marks II Optimal % Betting You play a sequence of Bernoulli games with the chance of you winning = p >1/2. (Why must his be?) You start with a stash of $D. If you bet and loose, you loose what you bet. If you win, your bet is matched. What is the optimal % bet for steady state winning?

3. copyright Robert J. Marks II Optimal % Betting Example Sequence –WinD[1]= D + %D =(1+%)D –Loose D[2]=D[1]- %D[1]= (1+%) (1-%)D –Loose D[3]=D[2]- % D[2]= (1+%) (1-%) 2 D –Win D[4]= (1+%) 2 (1-%) 2 D n trials and k wins leaves a stash of D[n]=(1+%) k (1-%) n-k D

4. copyright Robert J. Marks II Optimal % Betting n trials and k wins leaves a stash of D[n]=(1+%) k (1-%) n-k D What value of % maximizes D[n] for large n? Note: Maximizing D[n] for a fixed n is the same as maximizing

5. copyright Robert J. Marks II Optimal % Betting The law of large numbers says and Thus

6. copyright Robert J. Marks II Optimal % Betting Solving The optimal return is then D[n] = (1+%) k (1-%) n-k D = 2 n p k (1-p) n-k D Since Sanity Check: 100% for p=1 0% for p=1/2

7. copyright Robert J. Marks II Optimal % Betting The number of turns to double your stash is obtained by setting this to two and solving for n. The result is

8. copyright Robert J. Marks II Optimal % Betting p Notes: At p=1/2, n double =  At p=1, n double = 1

9. copyright Robert J. Marks II Optimal % Betting p

10. copyright Robert J. Marks II Roulette p = even or green = 20/36 = 0.5557 % = 2p-1 = 11.13% n double = 111.655

11. copyright Robert J. Marks II What if there are odds? Extend the problem to where you loose % of your bet when you loose and get w% of your bet when you win. Either or w can exceed one. What is the relation among w,,and p to assure you are in a position to win. Compute the optimal % of your bet. Optimal % Betting