Risk of Ruin

Q: Kya progression risk of ruin kam karti hai?

Martingale chhote sessions mein ise kam karti hai aur lambe sessions mein bahut zyada badha deti hai. Bade unit par flat betting ise kam karne ka imaandaar tareeka hai.

Target tak pahunchne se pehle poora bankroll haar jaane ki probability.

Chhota jawab

Even-money bets ke liye risk of ruin gambler’s ruin formula follow karta hai. $500 bankroll, $10 bets aur $600 target ke saath, aap lagbhag 57% baar target tak pahunchte hain aur baaki baar bust ho jaate hain.

formula

gambler’s ruin

p (even money)

18/37

q/p

1.0556

sabse purana result

Pascal–Fermat, 1656

Sawaal

Aapke paas a unitspaisaUnitBase stake jismein koi system ya player kaam karta hai. Progressions units mein batayi jaati hain taaki kisi bhi table par laagu hon.Poora padhein → ka bankrollpaisaBankrollKhelne ke liye alag rakhi gayi kul raqam. Bankroll management ka matlab hai tay karna ki har bet aur har session mein iska kitna hissa risk karna hai.Poora padhein → hai aur aap a + b units tak pahunchna chahte hain, ek baar mein ek unit ek even-money betbetEven-money betRed/black, odd/even, high/low. 18 numbers, payout 1:1.Poora padhein → par lagakar, jiski jeet ki probabilityganitProbabilityKisi outcome ka chance, 0 se 1 tak ya 0% se 100% tak. Ek number: 1/37 = 2.70%.Poora padhein → p = 18/37 aur haar ki q = 19/37 hai. Zero tak pahunchne se pehle target tak pahunchne ka chance kya hai? Iska jawab 1650s mein Pascal, Fermat aur Huygens ne nikala tha aur yeh probability ke sabse purane results mein se ek hai.

Blaise Pascal: Roulette Wheel ki Kahani

Gambler’s ruin formula

P(target tak pahunchna) = (1 − (q/p)ᵃ) / (1 − (q/p)ᵃ⁺ᵇ). Risk of ruinganitRisk of ruinTarget ya spin count tak pahunchne se pehle poora bankroll haar jaane ki probability.Poora padhein → ek minus yeh hai. q/p = 19/18 ≈ 1.0556 ke saath ratio units ki ginti ke saath exponentially badhta hai, isliye formula chhote units ko sakhti se saza deta hai: target tak pahunchne mein jitne zyada bets lagte hain, raaste mein utni baar 2.70% ka toll chukana padta hai.

bankrollunittargettarget tak betsp(target tak pahunchna)risk of ruin

$500$10$60010≈ 57%≈ 43%

$500$50$6002≈ 79%≈ 21%

$500$10$1,00050≈ 6%≈ 94%

$500$100$1,0005≈ 43%≈ 57%

$500$500$1,000148.6%51.4%

$100$10$20010≈ 37%≈ 63%

Even-money bets, flat stake, la partage nahi. Risk of Ruin Calculator kisi bhi numbers ka hisaab karta hai.

Ise padhna

Chhote units nuksaan karte hain. $10 bets se $500 ko double karne ki koshish 6% baar safal hoti hai; ek single $500 bet se 48.6% baar. Bold play, yaani kam aur bade bets, negative-expectation game mein target tak pahunchne ka chance sabse zyada karta hai, kyunki yeh edge laagu hone ki ginti sabse kam karta hai. Yeh aapke saath jo hota hai uska varianceganitVarianceYeh naap ki results expected value ke aas-paas kitna faile hue hain. Straight-up ke liye zyada, even-money bets ke liye kam.Poora padhein → bhi sabse zyada karta hai, aur shaam jaldi khatam kar deta hai. Is theorem ko “bold play is optimal” kehte hain aur ise Dubins aur Savage ne 1965 mein prove kiya.

Red par $500 ko $1,000 banane ka chance%

ek $500 bet

48.6%

$100 bets

≈ 43%

$50 bets

≈ 37%

$20 bets

≈ 21%

$10 bets

≈ 6%

$5 bets

≈ 0.5%

024.3048.6%

Jitne zyada bets lagte hain, utni zyada baar edge chukaya jaata hai. Darpok khel mehnga padta hai.

Problem kahan se shuru hui

Yeh sawaal roulette se purana hai. 1656 mein Pascal ne Fermat ko ek paheli di: do players teen dice phenkte hain, dono 12 counters se shuru karte hain; 11 aane par ek counter ek taraf jaata hai, 14 aane par doosri taraf. Dono ne ise hal kiya, aur Huygens ne, jinhe mathematician Pierre de Carcavi ki ek chitthi se iska pata chala, 1657 mein ise probability ki pehli kitaab ki paanchvi problem ke roop mein jawab ke saath aur bina proof ke chhaapa. Gyaarah 27 tarah se aa sakta hai aur chaudah sirf 15 tarah se, isliye har counter 14 mein 9 baar pehle player ki taraf jaata hai, aur upar wala formula, har ek ke 12 counters ke saath, use sab kuch jeetne ki sab kuch haarne se lagbhag 1,157 guna zyada sambhavna deta hai. 2.70% edge kahin chhota hai, lekin wahi formula dikhata hai ki bahut saare bets mein woh kya karta hai.

Pascal ki paheli mein teen dice ka har throw players ke beech ek counter khiskata hai.

Ek asli bold bet

April 2004 mein Ashley Revell, ek naujawan Englishman, ne apna lagbhag sab kuch bech diya, kapde bhi, aur $1,35,300 lekar Las Vegas ke Plaza gaye. Kuch chhote bets ke baad unhone poori raqam ek baar redbetRed18 red numbers par even-money bet. French mein: rouge.Poora padhein → par laga di. BallsaamanBallChhoti ball jo rotor ki ulti disha mein track par chhodi jaati hai. Aaj ki balls ivorine ya Teflon ki hoti hain, 18–21 mm.Poora padhein → red 7 par giri aur woh $2,70,600 lekar nikle. Double-zerowheelZeroAkela green pocket jise koi even-money, dozen ya column bet cover nahi karta; house edge ka source. Doosre naam: single zero, 0.Poora padhein → wheel par woh single spin 47.37% ka chance tha aur average mein unhe $7,121 ka pada, ek bet par 5.26% edge. $10 bets mein baantne par wahi target lagbhag na-mumkin hota. Bold play bet ko achha nahi banata; use jitna ho sake chhota banata hai.

Ashley Revell ka poora bankroll double-zero wheel ke ek single spin par laga hai.

Bina target ke risk of ruin

Agar aap bas tab tak khelte hain jab tak bust na ho jaayein ya N spins ke baad ruk jaate hain, to sawaal ban jaata hai “N spins ke andar mera bankroll zero ko chhoone ka chance kya hai?”. General N ke liye koi closed form nahi hai lekin is site ke simulators ise nikalte hain: red par $10 ke saath $500 ka bankroll 100 spins mein lagbhag kabhi bust nahi hota (1% se kam), 500 spins mein lagbhag 9% baar bust hota hai, aur 5,000 mein lagbhag 95% baar. Koi stop rule na ho to probability 100% ki taraf jaati hai, jo house edgeganitHouse edgeHar stake ka woh hissa jo casino average mein rakh leta hai: European roulette mein 2.70%.Poora padhein → ka long-run version hai.

Das sessions, $500 bankroll, red par $25, 200 spinsresult vs shuruaat

1,000 sessions × 500 spins · violet p10/p25/p75/p90, light blue median, 81 halke samples · kisi spin ke quantiles dekhne ke liye chart par cursor le jaiye

p10 −$500
p25 −$500
median −$500
p75 +$0
p90 +$350

1,000 sessions × 500 spins · y-axis 1st–99th percentile tak kata hua (extremes −$500 se +$1,500) · 500 spins · $25 base · even money

Jo lines neeche chhoo leti hain, woh ruin ho chuki hain. Chhote unit ke saath woh zyada der chaltin aur usi jagah khatam hotin.

Long-Run Convergence: Roulette mein Law of Large Numbers

Kitni der, aur kitna aage

Do aur classical results woh sawaal hal karte hain jo players sach mein poochhte hain. Bina target ke, flat even-money bets European wheel par bankroll ke har unit ke liye average mein 37 spins chalte hain aur American par 19, kyunki har spin average mein ek unit ka 1/37 leta hai (American wheel par 1/19): 100 units average mein 3,700 spins chalte hain, haalanki aise aadhe sessions 3,134 ke andar khatam ho jaate hain (Session Length and Cost dekhiye). Aur bankroll kitna bhi bada ho, European wheel par kabhi bhi k units aage hone ka chance (18/19)ᵏ hai. Ek unit ke liye yeh 95% hai aur das ke liye 58%, lekin koi bhi bankroll, kitna bhi gehra ho, kabhi 13 units upar hone ka aadha chance nahi deta; American wheel par chance (9/10)ᵏ hai aur 7 units par aadhe se neeche gir jaata hai.

units aage, kisi bhi point parEuropeanAmerican

194.7%90.0%

576.3%59.0%

1058.2%34.9%

1349.5%25.4%

2033.9%12.2%

506.7%0.5%

1000.45%0.003%

Even-money bets, flat stake, itna bada bankroll ki kabhi khatam na ho. Single-zero wheel par chance (18/19)ᵏ hai aur double-zero par (9/10)ᵏ.

Session ki Lambai aur Kharcha American Roulette Wheel aur Rules

Fifty-fifty chance ke liye bankroll

Sawaal ulta kijiye: sab kuch haarne se pehle ek tay raqam jeetne ka kam se kam fifty-fifty chance paane ke liye aapko kitna laana hoga? Chhoti jeet ke liye zyada nahi. Badi jeet ke liye zaroori bankroll tezi se badhta hai, aur ek point ke baad koi bhi bankroll kaafi nahi hota, kyunki ek anant bankroll bhi kabhi utna aage pahunchne ka fifty-fifty se kam chance deta hai. Taylor (2021) ise aise kehte hain: American wheel par 100 units tak pahunchne ka fifty-fifty chance paane ke liye aapko 94 se shuru karna hoga, yaani 6 jeetne ke liye 94 ka risk. European wheel par 88 chahiye.

jeetne ke liyeEuropeanAmerican

1 unit22

2 units33

3 units45

4 units68

5 units712

6 units921

7 units12koi bhi bankroll kaafi nahi

8 units15koi bhi bankroll kaafi nahi

10 units24koi bhi bankroll kaafi nahi

12 units46koi bhi bankroll kaafi nahi

13 units ya zyadakoi bhi bankroll kaafi nahikoi bhi bankroll kaafi nahi

Units mein sabse chhota bankroll jo flat even-money bets ke saath bankroll haarne se pehle raqam jeetne ka kam se kam fifty-fifty chance deta hai. Upar wale formula se hamari calculation.

Practical seekh

Tay kijiye ki aap karna kya chahte hain. Agar goal ek khaas jeet hai, to bade aur kam bets lagaiye. Agar goal ek lambi shaam hai, to chhote bets lagaiye aur maan lijiye ki edge ek bade turnoverpaisaTurnoverSession ya muddat mein lagayi gayi kul raqam, dobara lagayi gayi jeet milakar. Doosre naam: handle, action.Poora padhein → ka apna 2.70% le lega. Agar goal dono hain, to kuch chhodna padega, aur aam taur par woh target hota hai.

Formula

P(target) = (1 − (q/p)ᵃ) / (1 − (q/p)ᵃ⁺ᵇ)

risk of ruin = 1 − P(target)

even money ke liye q/p = 19/18

ruin tak average spins, bina target = 37 × units

P(kabhi k units aage) = (18/19)ᵏ

Sawaal

Kya progression risk of ruin kam karti hai?

MartingalestrategyMartingaleProgression jo har haar ke baad bet double karti hai. Yeh kya karti hai aur kya nahi, iske liye Strategy section dekhiye.Poora padhein → chhote sessions mein ise kam karti hai aur lambe sessions mein bahut zyada badha deti hai. Bade unit par flat bettingstrategyFlat bettingHar spin par same raqam lagana. Har progression ko isi baseline se compare kiya jaata hai.Poora padhein → ise kam karne ka imaandaar tareeka hai.

Aur straight-up bets?

Upar wala formula even money ke liye hai. High-variance bets ke liye chhote stakes par ruin zyada likely hai aur target oonche-neeche tareeke se aata hai; calculator use kijiye.

q/p hi key quantity kyun hai?

Target ki taraf har kadam ruin ki taraf har kadam se thoda kam likely hai; yeh ratio kadmon ki ginti par compound hota hai.

Maths ke saath mazaa

Pehle khud try kijiye. Har solution chhupa hua hai; jab taiyaar hon tab kholiye.

Aapke paas $100 hain aur aapko $1,000 chahiye, sirf redbetRed18 red numbers par even-money bet. French mein: rouge.Poora padhein → par bet lagakar. Plan ek: ek baar mein $1 lagaiye. Plan do: jab tak $500 paar na ho, jo hai sab lagaiye, phir sirf utna jitna $1,000 tak pahunchne ke liye chahiye. Kaunsa plan behtar hai, aur kitna?
Solution dikhaiye
Jawab Plan do, bahut zyada. Yeh lagbhag 9.0% baar $1,000 tak pahunchta hai, fair 10% ke kareeb. Plan ek 10²² mein lagbhag 7 baar safal hota hai.
Fair gameganitFair gameAisa game jismein kisi bhi side ka edge nahi: har bet ki expected value zero hai.Poora padhein → mein aap $100 se $1,000 tak das mein ek baar pahunchte, chahe kuch bhi karte, kyunki aapke paise ki average value $100 hi rehti hai.
Edge har bet ka 2.70% leta hai, isliye aap total mein jitna kam lagayein, das mein ek ke utne kareeb rehte hain. Plan do mamla kuch hi bets mein tay kar deta hai: $100 se $200, $400, $800, aur phir $200 kaafi hai. Yeh lagbhag 9.0% baar safal hota hai. Plan ek mein walk khatam hone se pehle hazaaron bets lagti hain, aur edge, har ek par chhota, jud kar lagbhag pakki barbaadi ban jaata hai.
Yeh Dubins–Savage result ka sabse saral roop hai (Epstein, 2013): jo game house ke favour mein ho, usmein jitna chahiye utna lagaiye, jitni kam baar ho sake.
Aapke paas $500 hain aur aapko $1,000 chahiye, mil jaate hi ruk jaana hai. Aap ek tarah ki bet chunte hain aur use tab tak dohraate hain jab tak woh ek baar jeet na jaaye ya paisa khatam na ho. Kaunsi tarah sabse achha chance deti hai?
Solution dikhaiye
Jawab Red par ek $500 ki bet: 48.6%. Paise ko chhoti bets mein baantne wala har plan isse bura karta hai.
Har stake aise chuniye ki ek jeet kaafi ho: straight-upbetStraight-upEk single number, payout 35:1.Poora padhein → par $30 $1,050 jeetta hai, splitbetSplitDo saath wale numbers, payout 17:1.Poora padhein → par $60 $1,020 jeetta hai, aur isi tarah. Phir chance giniye ki paisa khatam hone se pehle kam se kam ek bet jeete. Solah $30 straight ups sab (36/37)¹⁶ = 64.5% chance se miss hote hain, isliye woh 35.5% baar $1,000 tak pahunchte hain.
Table har bet dikhati hai. Red sabse achha karta hai kyunki woh ek hi bet mein pahunch jaata hai, isliye edge ek hi baar, $500 par, lagta hai. Six linebetSix lineDo saath wali rows ke chhe numbers, payout 5:1. French mein: sixain.Poora padhein → doosre number par hai kyunki use sirf teen chahiye. Ek akeli dozenbetDozen1–12, 13–24 ya 25–36, payout 2:1.Poora padhein → bet sabse buri hai, 12/37 ka shot; $250 unitspaisaUnitBase stake jismein koi system ya player kaam karta hai. Progressions units mein batayi jaati hain taaki kisi bhi table par laagu hon.Poora padhein → mein $1,000 ya zyada hone tak khelne par bhi dozen sirf 46.4% baar pahunchta hai.
Bollman (2024) American wheel ke liye yahi table banate hain. Unki dozen row mein ek galti hai: wahan 12/38 = 31.6% hona chahiye.
betstake × bets ki sankhya$1,000 tak pahunchne ka chance
Straight-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%
Aap red par ek unit lagate hain, spin ke baad spin, aise bankrollpaisaBankrollKhelne ke liye alag rakhi gayi kul raqam. Bankroll management ka matlab hai tay karna ki har bet aur har session mein iska kitna hissa risk karna hai.Poora padhein → ke saath jo kabhi khatam nahi hota. Pehle spin ke baad aap ek unit upar ya ek unit neeche hain. Iska chance kya hai ki aap kabhi phir se bilkul barabar honge?
Solution dikhaiye
Jawab 37 mein 36, 97.3%. American wheel par yeh 38 mein 36, 94.7% hai.
Agar pehla spin jeetta hai (37 mein 18 baar), aap ek unit upar hain, aur edge har kabhi na khatam hone wale session ko neeche kheenchta hai, isliye der-saber aap barabar se guzarte hain: yeh pakka hai. Agar haarta hai (37 mein 19 baar), aapko ek unit chadhna hai, aur kabhi ek unit aage hone ka chance 18/19 hai (upar ki table dekhiye).
Milakar: 18/37 + 19/37 × 18/19 = 18/37 + 18/37 = 36/37.
Aam taur par chance hai 1 minus spin jeetne aur haarne ke chances ka antar: European wheel par 1 − 1/37, American par 1 − 2/38. Bina zerowheelZeroAkela green pocket jise koi even-money, dozen ya column bet cover nahi karta; house edge ka source. Doosre naam: single zero, 0.Poora padhein → wale game mein antar kuch nahi aur chance 1 hai: score hamesha barabar par laut-ta hai. Epstein (2013) general rule deta hai.
Pascal ki 1656 ki paheli: do khiladi teen dice phenkte hain, har ek 12 counters se shuru karta hai. Har 11 ek counter pehle khiladi ko deta hai, har 14 ek doosre ko; baaki throws nahi gine jaate. Pehle khiladi ke sabhi 24 jeetne ka chance kitna hai? Aur agar aap iske badle ek dost ke khilaaf, jo bank bana hai, har spin red par 1 unit lagayein, dono ke 12 units, to aapke sabhi 24 jeetne ka chance kitna hai?
Solution dikhaiye
Jawab Pehla khiladi lagbhag 1,158 mein 1,157 baar jeetta hai. Aap, red par, sabhi 24 units sirf lagbhag 34% baar jeette hain.
Teen dice 27 tareekon se 11 aur 15 tareekon se 14 banate hain, isliye har hilne wala counter 42 mein 27 baar, yaani 14 mein 9 baar, pehle khiladi ko jaata hai. Dono taraf 12 se shuru karne par sab kuch jeetne ka chance 1/(1 + r¹²) hai, jahan r doosri taraf ke chance ka aapke chance se ratio hai.
Pehle khiladi ke liye r = 5/9, aur (5/9)¹² ≈ 0.00086, isliye chance lagbhag 0.99914 hai: 9¹² to 5¹² ke odds, yaani 2,82,42,95,36,481 to 24,41,40,625, lagbhag 1,157 to 1. Huygens ne 1657 mein theek yahi numbers chhaape.
Red par r = 19/18 hai zero ki wajah se, aur (19/18)¹² ≈ 1.913: aapka chance 1/2.913 ≈ 34.3% hai. Ek coin 50% deta. Har bet par 2.70% edge, jitni bahut saari bets lagti hain unpar jud kar, 16-point ka antar ban jaata hai.

Thodi hansi

$Bold play optimal hai. Yeh jaldi khatam bhi ho jaata hai.$

Bold play optimal hai. Yeh jaldi khatam bhi ho jaata hai.

Sources

Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 1. Wiley, 3rd ed., 1968. Gambler’s ruin, chapter XIV.
Epstein, R. A. The Theory of Gambling and Statistical Logic, 2nd ed. Academic Press, 2013.
Taylor, D. G. Games, Gambling, and Probability: An Introduction to Mathematics, 2nd ed. CRC Press, 2021.
Carcavi, P. de. Letter to Huygens, 28 September 1656, on the problem Pascal posed to Fermat. Œuvres complètes de Christiaan Huygens, vol. 1, 1888, pp. 492–494.
Huygens, C. De Ratiociniis in Ludo Aleae. 1657. Expectation ka sabse pehla chhapa hua vivaran.
Ethier, S. N. The Doctrine of Chances: Probabilistic Aspects of Gambling. Springer, 2010. Chapter 13, roulette.
Reuters. “Briton bets it all on roulette spin, and wins.” April 2004.
Las Vegas Sun. “Man wagers life’s worth.” 12 April 2004.

2026-09-28 ko review kiyamodel: European, single zero walamethodology ↗sources ↗