We need a rule clarification from OP:
Suppose you bet $10 on Red and Red it is; the croupier hands you $20 total. Does (a) the entire $20 go in your left pocket, or (b) just $10 goes in the left pocket with your $10 bet staying on the table or in your right pocket?
For this post I’ll assume the easy case (a). Case (b) is more interesting and I may consider it in a follow-up.
It sounds like OP does NOT want to consider enjoyment (e.g. deliberately prolonging the action for fun), and only wants to maximize the expectation of left pocket. As Chronos implies, the expectation will be precisely $1000 minus the house vig, or $947 in the case of roulette. (That $947 is achieved most easily by simply betting all at once $26.31 on each of 38 numbers.)
In the 20th century you could expect $995 (or more depending on rules) playing Blackjack, even without counting cards provided that you used proper strategy. (I’m not sure about now. Are all the casinos now using less favorable rules?)
Perhaps more interesting than the mean amount that ends up in your pocket, is the variance of your winnings! To maximize variance at roulette, bet the entire $1000 on Number 18[sup]*[/sup] — you’ll have a 2.632% chance of walking away with $36,000.
-
- “Today’s my birthday. I’m not a prime anymore.”