Say there was a game like where you rolled 2 6-sided dice and made a custom dollar bet based on if you thought the next roll would be higher or lower. The game had the following conditions:
- You start with $1000 in your account
- Each round you bet whether you think the next die roll will be higher or lower than the last one
- In case of a tie your bet carries over to the next round
- You can bet any amount in your account, including 0
- If you lose all your money, the game is over
- The game is over after 10 rolls. Whatever is in your account at that time you get to keep.
What would be the optimum betting strategy so you ended up with the most money at the end? The obvious bets are when the current dice are showing 1 or 12, then bet it all on higher or lower. But what about when the current dice are showing 2-11? If you just bet straight odds and lose a couple of rounds, you could end up losing a big chunk of your money. Some of the scenarios I can think of:
No bet: Always bet $0 and go home with 1000
Obvious bet: Only bet on the 1 or 12. Most likely go home with ??? (minimum $1000)
Odds bet: Based on the high low probability, bet P and most likely go home with ???
What would the optimum betting strategy be in a game like this?