Roulette table

A 100-chip dozen bet offers a 200-chip gain on 12 of 38 pockets, providing meaningful progress toward the 500-chip shortfall while preserving bankroll for the remaining spins. All dozens have identical probabilities on this independent, uniform wheel.

A win raises the bankroll to 1,530, meeting the target immediately. The small stake preserves chips for further attempts if this spin loses; every pocket is equally likely.

Covering eight pockets gives an 8/38 chance of reaching 1,526 chips immediately. A miss leaves 698 chips and 17 spins to pursue the target.

Any of these seven hits raises the bankroll to 1,510, immediately meeting the target with a 7/38 probability. The 196-chip total stake leaves 502 chips for further attempts if this spin misses.

A win raises the bankroll to 1,517 chips, reaching the target immediately. This is the smallest straight-up stake that achieves that, preserving chips for further attempts if it loses.

A 30-chip straight-up win raises the bankroll to 1,523, meeting the target immediately while preserving chips for further attempts if it loses. Every pocket is equally likely; previous outcomes do not affect this spin.

A 31-chip straight-up win would raise the bankroll to 1,528, reaching the target immediately. This is the smallest single straight-up stake that does so, preserving chips for further attempts if it loses.

A 32-chip straight-up win raises the bankroll to 1,532, meeting the target immediately. This is the smallest stake that can reach the target in one win, preserving chips for further attempts if it loses.

A 32-chip straight-up win reaches exactly 1,500 chips, while a loss preserves most of the bankroll for further attempts. All pockets are equally likely, so the choice of 0 does not rely on previous outcomes.

A 33-chip straight-up win raises the bankroll to 1,503, reaching the target immediately. This is the smallest straight-up stake that achieves the goal while preserving chips for further attempts if it loses.

A win raises the bankroll to 1,505, reaching the target. This is the smallest straight-up stake that does so, preserving chips for further attempts if it loses.

Any of these four hits brings the bankroll to 1,529, meeting the target immediately. If all miss, 125 chips remain for further attempts during the eight remaining spins.

A 40-chip straight-up win raises my bankroll to 1,525, reaching the target immediately. It also preserves 85 chips after a loss for further attempts; every pocket is equally likely.

A hit raises the bankroll to 1,520 chips, meeting the target. A miss leaves 44 chips, enough for another direct attempt with spins remaining.

A 42-chip straight-up win raises my bankroll to 1,514, immediately meeting the target. This is the smallest stake that can reach the target in one spin, preserving two chips if it loses.

With only five spins left, staking both chips on a straight-up bet offers a chance to reach 72 chips, enough to make a target-reaching wager next spin. Lower-payout bets would require more successful spins to reach 1,500.

The bankroll is zero and no bets are legal, so skipping is the only available action.

With a bankroll of zero and no legal bets available, skipping is the only legal action.

The bankroll is zero and no bets are legal, so I must skip this spin.

The bankroll is zero and no bets are legal, so the only valid action is to place no bets.