Quote:
Originally Posted by David Sklansky
Suppose the goal was to get one dollar ahead.
This is pretty complicated and I don't think I can solve it without a computer.
I think the starting point is that we always want to have exactly 10.5 on the final 10th roll no matter what if we haven't already reached $21. So we can work backwards from there.
10th roll:
Bankroll $10.5 Bet $10.5 [50%]
9th roll:
Bankroll $15.75 Bet $5.25 [75%] -> this is simple
Bankroll $10.5 Stand Pat [50%] -> we bet $0 here to gain information
Bankroll $5.25 Bet $5.25 [25%] -> we need to double up twice
Here it gets more interesting:
8th roll:
Bankroll $18.375 Bet $2.625 [87.5%] -> WIN or $15.75
Bankroll $13.125 Bet $2.625 [62.5%] -> $15.75 or $10.5
or
Bankroll $13.125 Bet $7.875 [62.5%] -> WIN or $5.25 (I don't think this is ever correct)
Bankroll $10.5 Bet $5.25 [50%]-> $15.75 or $5.25
I'm not sure of all the possible states here without a more complete solution. Other possibilities on 8th roll:
Bankroll $7.875 Bet $2.625 [37.5%] -> $10.5 or $5.25 or All in -> $15.75 or BUSTO [37.5%]
Bankroll $5.25 [25%] Stand pat.
Bankroll $2.625 [12.5%] All in.
Obviously it blows up from here when we get to the 7th step and I'd need to use a computer to solve this. If I'm feeling ambitious I might work on it this weekend.