Risk a ruin
Di probability fi lose yuh whole bankroll bifo yuh reach a target.
Fi even-money bet, risk a ruin follow di gambla ruin formula. Wid a $500 bankroll, $10 bet an a $600 target, yuh reach di target bout 57% a di time an bus out di res.
Di question
Yuh have a unitmoney mattaUnitDi base stake weh a system or a playa work wid. Dem describe progression inna unit suh dem work at any table.Read di whole ting → a bankrollmoney mattaBankrollAll di money yuh set aside fi play. Fi manage yuh bankroll mean yuh decide ow much a it yuh a go risk pon evry bet an evry session.Read di whole ting → an yuh waan reach a + b unit, an yuh bet one unit at a time pon a even-money betbet demEven-money betRed/black, odd/even, high/low. 18 numba, pay 1:1.Read di whole ting → wid win probabilitymathsProbabilityDi chance fi a result, fram 0 to 1 or 0% to 100%. One numba: 1/37 = 2.70%.Read di whole ting → p = 18/37 an loss probability q = 19/37. Wah di chance seh yuh reach di target bifo yuh reach zero? Pascal, Fermat an Huygens work out di ansa inna di 1650s, an it a one a di oldes result inna probability.
Di gambla ruin formula
P(reach target) = (1 − (q/p)ᵃ) / (1 − (q/p)ᵃ⁺ᵇ). Risk a ruinmathsRisk a ruinDi probability fi lose di whole bankroll bifo yuh reach a target or a spin count.Read di whole ting → a one minus dat. Wid q/p = 19/18 ≈ 1.0556, di ratio grow exponential wid di count a unit, suh di formula punish small unit hard: di more bet it tek fi reach di target, di more time yuh pay di 2.70% toll pon di way.
Ow fi read it
Smalla unit hurt. Fi try double $500 wid $10 bet work 6% a di time; wid one single $500 bet it work 48.6% a di time. Bold play, fewa an bigga bet, give di bes chance fi hit a target inna a game wid negative expectation, cause it cut dung di count a time di edge apply. It also give di bigges variancemathsVarianceA measure a ow wide di result dem spread roun di expected value. High fi straight up, low fi even-money bet.Read di whole ting → fi wah happen to yuh, an it end di evenin quick. Di theorem name “bold play is optimal” an Dubins an Savage prove it inna 1965.
Weh di problem start
Di question older dan roulette. Inna 1656 Pascal set Fermat a puzzle: two playa trow tree dice, each one start wid 12 counter; a trow a 11 move a counter one way, a trow a 14 move it di odda way. Both a dem solve it, an Huygens, weh hear bout it tru a letta fram di mathematician Pierre de Carcavi, print it inna 1657 as di fifth problem inna di fus book pon probability, wid di ansa an no proof. Yuh can trow eleven 27 way an fourteen only 15, suh each counter move di fus playa way 9 time outta 14, an di formula up top, wid 12 counter each, mek him bout 1,157 time more likely fi win evryting dan fi lose evryting. A 2.70% edge much smalla, but di same formula show wah it do ova nuff bet.
A bold bet fi real
Inna April 2004 Ashley Revell, a young Englishman, sell nearly evryting him own, even him clothes, an tek $135,300 to di Plaza inna Las Vegas. Afta some small bet him put di whole amount pon redbet demRedEven-money bet pon di 18 red numba. Inna French: rouge.Read di whole ting →, one time. Di ballgearBallDi likkle ball weh di dealer fling roun di track di opposite way to di rotor. Modern ball mek outta ivorine or Teflon, 18–21 mm.Read di whole ting → land pon red 7 an him walk out wid $270,600. Pon a double-zeropon di wheelZeroDi one green pocket weh no even-money, dozen or column bet nuh cover; a weh di house edge come fram. Nex name: single zero, 0.Read di whole ting → wheel, dat one spin did a 47.37% chance an it cost him $7,121 pon average, di 5.26% edge pon one bet. Spread ova $10 bet, di same target woulda nearly hopeless. Bold play nuh mek di bet good; it mek it as short as it can go.
Risk a ruin widout a target
If yuh jus play till yuh bus or stop afta N spin, di question tun inna “wah di chance seh mi bankroll touch zero inside N spin?”. No closed form nuh deh fi general N, but di simulator dem pon dis site work it out: a $500 bankroll at $10 pon red almost neva bus inna 100 spin (unda 1%), bus bout 9% a di time inna 500 spin, an bout 95% a di time inna 5,000. Wid no stop rule, di probability head fi 100%, an dat a di long-run version a di house edgemathsHouse edgeDi part a evry stake weh di casino keep pon average: 2.70% inna European roulette.Read di whole ting →.
- p10 −$500
- p25 −$500
- median −$500
- p75 +$0
- p90 +$350
Ow long, an ow far ahead
Two more classic result ansa question weh playa really ask. Wid no target, flat even-money bet las pon average 37 spin fi evry unit a bankroll pon a European wheel an 19 pon a American one, cause each spin tek 1/37 a a unit pon average (1/19 pon a American wheel): 100 unit las 3,700 spin pon average, even though half a dem session done inside 3,134 (see Session Length and Cost). An no matta ow big di bankroll, di chance pon a European wheel fi eva get k unit ahead a (18/19)ᵏ. Dat a 95% fi one unit an 58% fi ten, but no bankroll, no matta ow deep, nuh give a even chance fi eva get 13 unit up; pon a American wheel di chance a (9/10)ᵏ an it drop unda one half at 7 unit.
Ow long di session las an wah it costAmerican roulette: di wheel an di rule dem
A bankroll fi a fifty-fifty chance
Turn di question roun: ow much yuh haffi bring fi have at least a fifty-fifty chance fi win a set amount bifo yuh lose evryting? Fi a small win, nuh much. Fi a bigga one, di bankroll yuh need climb steep, an afta a point no bankroll nuh enough, cause even one weh nuh have no end give less dan a fifty-fifty chance fi eva get dat far ahead. Taylor (2021) put it anodda way: pon a American wheel, fi have a fifty-fifty chance fi reach 100 unit, yuh haffi start wid 94, an risk 94 fi win 6. Pon a European wheel yuh woulda need 88.
Di practical lesson
Decide wah yuh a try fi do. If di goal a one set win, bet big an few. If di goal a a long evenin, bet small an accep seh di edge wi tek it 2.70% outta a big turnovermoney mattaTurnoverAll di money yuh bet ova a session or a period, countin di winnins yuh bet again (all di money yuh bet, roun afta roun). Nex name: handle, action.Read di whole ting →. If di goal a both, sumting haffi give, an most time a di target.
Formula
Question dem
Martingalestrategy demMartingaleProgression weh double di bet afta evry loss. Look inna di Strategy section fi see wah it do an wah it nuh do.Read di whole ting → bring it dung fi short session an push it up bad fi long one. Flat bettinstrategy demFlat bettinFi bet di same amount evry spin. Di baseline weh dem compare evry progression gainst.Read di whole ting → at a bigga unit a di honest way fi bring it dung.
Di formula up top a fi even money. Fi high-variance bet, ruin more likely at small stake an yuh reach di target inna a more lumpy way; use di calculator.
Each step toward di target a likkle less likely dan each step toward ruin; di ratio compound ova di count a step.
Fun wid maths
Try dem yuhself fus. Evry ansa fold up; open it wen yuh ready.
Yuh have $100 an yuh waan $1,000, an yuh bet only pon redbet demRedEven-money bet pon di 18 red numba. Inna French: rouge.Read di whole ting →. Plan one: bet $1 at a time. Plan two: bet evryting yuh have till yuh pass $500, den jus enough fi reach $1,000. Which plan betta, an by ow much?
Show di ansa
Ansa Plan two, by far. It reach $1,000 bout 9.0% a di time, close to di fair 10%. Plan one work bout 7 time outta 10²².
Inna a fair gamemathsFair gameA game weh nuh side nuh have no edge: evry bet have a expected value a zero.Read di whole ting → yuh woulda reach $1,000 fram $100 one time outta ten no matta wah yuh do, cause di average value a yuh money stay at $100.
Di edge tek 2.70% a evry bet, suh di less yuh bet inna total, di closer yuh stay to one outta ten. Plan two settle di matta inna a handful a bet: $100 turn $200, $400, $800, an den $200 a enough. It work bout 9.0% a di time. Plan one tek tousands a bet bifo di walk end, an di edge, small pon each one a dem, add up to near sure ruin.
Dis a di Dubins–Savage result inna it simplest form (Epstein, 2013): inna a game weh favour di house, bet as much as yuh need, as few time as yuh can.
Yuh have $500 an yuh waan $1,000, an yuh stop as soon as yuh have it. Yuh pick one kind a bet an repeat it till it win one time or di money run out. Which kind give di bes chance?
Show di ansa
Ansa One $500 bet pon red: 48.6%. Evry plan weh spread di money ova smalla bet do worse.
Choose each stake suh one win a enough: $30 pon a straight upbet demStraight upOne single numba, pay 35:1.Read di whole ting → win $1,050, $60 pon a splitbet demSplitTwo numba weh deh side by side, pay 17:1.Read di whole ting → win $1,020, an suh on. Den count di chance seh at least one bet win bifo di money run out. Sixteen $30 straight up all miss wid chance (36/37)¹⁶ = 64.5%, suh dem reach $1,000 35.5% a di time.
Di table show each bet. Red do bes cause it get deh inna one bet, suh di edge get pay one time, pon $500. Di six linebet demSix lineSix numba inna two row weh deh side by side, pay 5:1. Inna French: sixain.Read di whole ting → come second cause it need only tree. A single dozenbet demDozen1–12, 13–24 or 25–36, pay 2:1.Read di whole ting → bet a di worse, a 12/37 shot; even if yuh play it inna $250 unitmoney mattaUnitDi base stake weh a system or a playa work wid. Dem describe progression inna unit suh dem work at any table.Read di whole ting → till yuh have $1,000 or more, di dozen get deh only 46.4% a di time.
Bollman (2024) build di same table fi a American wheel. Im dozen row have a mistake: it fi read 12/38 = 31.6%.
betstake × ow much betchance fi reach $1,000Straight up$30 × 1635.5%Split$60 × 835.9%Street$90 × 534.5%Corner$125 × 436.7%Six line$166, $166, $16841.2%Dozen$500 × 132.4%Red$500 × 148.6%Yuh bet one unit pon red, spin afta spin, wid a bankrollmoney mattaBankrollAll di money yuh set aside fi play. Fi manage yuh bankroll mean yuh decide ow much a it yuh a go risk pon evry bet an evry session.Read di whole ting → weh neva run out. Afta di fus spin yuh one unit up or one unit dung. Wah a di chance seh yuh wi eva come back exactly even again?
Show di ansa
Ansa 36 outta 37, 97.3%. Pon a American wheel it a 36 outta 38, 94.7%.
If di fus spin win (18 time outta 37), yuh one unit up, an di edge drag dung evry session weh neva end, suh sooner or lata yuh pass back tru even: dat a sure. If it lose (19 time outta 37), yuh haffi climb one unit, an di chance fi eva reach one unit ahead a 18/19 (look pon di table up top).
Togedda: 18/37 + 19/37 × 18/19 = 18/37 + 18/37 = 36/37.
Inna general di chance a 1 minus di gap between di chance fi win an di chance fi lose a spin: 1 − 1/37 pon a European wheel, 1 − 2/38 pon a American one. Inna a game wid no zeropon di wheelZeroDi one green pocket weh no even-money, dozen or column bet nuh cover; a weh di house edge come fram. Nex name: single zero, 0.Read di whole ting → di gap a nutten an di chance a 1: di score always come back to even. Epstein (2013) give di general rule.
Pascal 1656 puzzle: two playa throw tree dice, an each one start wid 12 counter. Evry 11 move a counter to di fus playa, evry 14 move one to di second; odda throw nuh count. Ow likely di fus playa fi win all 24? An if yuh bet 1 unit pon red each spin instead, gainst a fren weh act as di bank, 12 unit each, ow likely yuh fi win all 24?
Show di ansa
Ansa Di fus playa win bout 1,157 time outta 1,158. Yuh, pon red, win all 24 unit only bout 34% a di time.
Tree dice mek 11 inna 27 way an 14 inna 15 way, suh each counter weh move go to di fus playa 27 time outta 42, 9 time outta 14. Wid each side a start pon 12, di chance fi tek evryting a 1/(1 + r¹²), weh r a di ratio a di odda side chance to fi yuh.
Fi di fus playa r = 5/9, an (5/9)¹² ≈ 0.00086, suh di chance a bout 0.99914: odds a 9¹² to 5¹², or 282,429,536,481 to 244,140,625, bout 1,157 to 1. Huygens print exactly dem numba deh inna 1657.
Pon red, r = 19/18 cause a di zero, an (19/18)¹² ≈ 1.913: yuh chance a 1/2.913 ≈ 34.3%. A coin woulda give 50%. Di 2.70% edge pon each bet add up, ova all di bet it tek, to a 16-point gap.
Sumting fi laugh
Bold play a di bes. It done quick too.
Source dem
- Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 1.
- Epstein, R. A. The Theory of Gambling and Statistical Logic, 2nd ed.
- Taylor, D. G. Games, Gambling, and Probability: An Introduction to Mathematics, 2nd ed.
- Carcavi, P. de. Letta to Huygens, 28 September 1656, pon di problem weh Pascal set fi Fermat.
- Huygens, C. De Ratiociniis in Ludo Aleae.
- Ethier, S. N. The Doctrine of Chances: Probabilistic Aspects of Gambling.
- Reuters. “Briton bets it all on roulette spin, and wins.”
- Las Vegas Sun. “Man wagers life’s worth.”