Roulette table

I chose a legal action for this decision.

These bets give an 8/38 chance of reaching 1,504 chips immediately, with little excess above the target, while retaining 856 chips after a miss. The selected pockets are arbitrary because spins are independent and uniform.

I chose a legal action for this decision.

Staking 184 across eight pockets gives an 8/38 chance of reaching exactly 1,500 immediately: 856 − 184 + 36×23 = 1,500. A miss leaves 672 chips and 18 spins for further attempts.

I chose a legal action for this decision.

This gives a 6/38 chance of reaching 1,512 chips immediately, while leaving 504 chips after a miss for the remaining 17 spins. The selected pockets are arbitrary because every spin is independent and uniform.

I chose a legal action for this decision.

A 29-chip straight-up win raises the bankroll to 1,519, clearing the target with the smallest single-number stake that can do so. A loss preserves 475 chips for the remaining spins; every pocket is equally likely.

I chose a legal action for this decision.

A 30-chip straight-up win brings the bankroll to 1,525, reaching the target while preserving chips for further attempts if it loses. Every pocket is equally likely, regardless of previous outcomes.

I chose a legal action for this decision.

A 31-chip straight-up win would raise my bankroll to 1,530, reaching the target immediately while preserving most chips if it misses. All pockets are equally likely; past outcomes do not affect this choice.

I chose a legal action for this decision.

Either hit reaches 1,502 chips, just above the target, while a miss preserves 350 chips for the remaining spins. Both pockets are equally likely; previous outcomes do not affect this spin.

I chose a legal action for this decision.

A 33-chip straight-up win reaches 1,505 chips, meeting the target with minimal overshoot while preserving 317 chips if it loses. Every pocket is equally likely; previous outcomes do not affect this choice.

I chose a legal action for this decision.

A 34-chip straight-up win brings the bankroll to 1,507, reaching the target with the smallest stake that can do so immediately. A loss preserves 283 chips for further attempts; every pocket is equally likely.

I chose a legal action for this decision.

A 35-chip straight-up win raises my bankroll to 1,508, meeting the target without unnecessary overshoot. This keeps 248 chips available after a loss; all pockets are equally likely regardless of previous outcomes.

I chose a legal action for this decision.

A 36-chip straight-up win brings the bankroll to 1,508, meeting the target without unnecessary overshoot. A loss leaves 212 chips for further attempts; all pockets are equally likely.

I chose a legal action for this decision.

A 37-chip straight-up win reaches 1,507 chips, meeting the target without unnecessary overshoot. A loss leaves 175 chips for the remaining eight spins; every pocket is equally likely.

I chose a legal action for this decision.

A 38-chip straight-up win reaches 1,505 chips, meeting the target without unnecessary overshoot. A loss leaves 137 chips and seven spins for further attempts; every pocket is equally likely.

I chose a legal action for this decision.

A 39-chip straight-up win reaches 1,502 chips, meeting the target immediately. This is the smallest stake that can do so, preserving 98 chips for further attempts if it loses.

I chose a legal action for this decision.

A 41-chip straight-up win reaches 1,533 chips, meeting the target immediately; a loss leaves 57 chips for the five remaining spins. All pockets are equally likely, so the choice of zero does not rely on past outcomes.

I chose a legal action for this decision.

A 42-chip straight-up win reaches 1,527 chips, clearing the target immediately. This is the smallest stake that does so, preserving 15 chips for the remaining four spins if it loses.

I chose a legal action for this decision.

With only four spins left, an all-in dozen bet gives a 12/38 chance to reach 45 chips, enough for a subsequent straight-up win to reach the target. All dozens are equally likely; past outcomes do not affect this choice.

I chose a legal action for this decision.

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

I chose a legal action for this decision.

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

I chose a legal action for this decision.

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