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Last Updated: April 18, 2016

Royal or Bust

Introduction



The purpose of this page is to answer such questions as, "If I have a bankroll of $2,000, what is my probability of hitting a royal flush before going broke?" The information in this page will not increase the player's odds, but it may be of use to the player in planning a budget if his goal is to hit a royal or go bust trying.

In addition to a "royal or bust" table, we will also look at the bankroll needed to hit four deuces or better in two deuces wild games. We also have a table for the probability of ruin by bankroll size if the goal is to hit a win of 25 or more, which is a four of a kind or better in non-wild card games and a wild royal or better in deuces wild games.

In all tables, the bankroll is measured in full bets. For example, a bankroll of 100 at a $1 game would be $500, because the player should be betting five coins, or in this case $5, at a time.

Games



Following is a list of games covered by this analysis.

Game Summary

Abbreviation Full Name Full Pay Table Royal Probability Return
9-6 DDB Double Double Bonus 1,1,3,4,6,9,50,80,160,160,400,50,800 1 in 40799 98.98%
9-6 JoB Jacks or Better 1,2,3,4,6,9,25,50,800 1 in 40391 99.54%
FPDW Full pay Deuces Wild 1,2,2,3,5,9,15,25,200,800 1 in 45282 100.76%
NSUD Not so ugly ducks (Deuces Wild) 1,2,3,4,4,10,16,25,200,800 1 in 43456 99.73%
10-7 DB Double Bonus 1,1,3,5,7,10,50,80,160,50,800 1 in 48048 100.17%
8-5 BP Bonus Poker 1,2,3,4,5,8,25,40,80,50,800 1 in 40233 99.17%
FPJ Full pay Joker Wild 1(kings),1,2,3,5,7,20,50,100,200,800 1 in 41214 100.65%
FPA8 Full pay Aces and Eights 1,2,3,4,5,8,25,50,80,50,800 1 in 40233 99.78%


Royal or Bust



It seems to me that recreational gamblers often have a weekend goal of hitting a royal, at which time they immediately quit playing. The following table strives to answer the question of the probability of success for various bankroll sizes and common games.

For example, consider a player has a bankroll of $2,000 and is playing a 9-6 Jacks or Better on a 25¢ game. A full bet, which video poker players should always make, is five quarters, or $1.25. That comes to a bankroll of $2,000/$1.25 = 1,600 bets. The table shows the player will have a 78.03% chance of hitting a royal before going bust.

Royal or Bust

Bankroll 9-6 DDB 9-6 JoB FPDW NSUD 10-7 DB 8-5 BP FPJ FPA8
100 6.26% 9.05% 12.08% 8.52% 9.10% 7.89% 11.72% 9.51%
200 12.12% 17.27% 22.73% 16.32% 17.34% 15.16% 22.13% 18.11%
300 17.62% 24.78% 32.08% 23.46% 24.86% 21.85% 31.29% 25.88%
400 22.78% 31.59% 40.25% 29.96% 31.70% 28.04% 39.35% 32.91%
500 27.62% 37.77% 47.50% 35.93% 37.93% 33.69% 46.51% 39.29%
600 32.15% 43.37% 53.84% 41.36% 43.56% 38.92% 52.82% 45.03%
700 36.40% 48.48% 59.41% 46.38% 48.69% 43.74% 58.39% 50.26%
800 40.38% 53.12% 64.34% 50.95% 53.38% 48.17% 63.27% 54.96%
900 44.10% 57.34% 68.64% 55.15% 57.61% 52.26% 67.57% 59.24%
1000 47.61% 61.21% 72.42% 58.99% 61.49% 56.03% 71.38% 63.12%
1100 50.89% 64.73% 75.76% 62.50% 65.00% 59.51% 74.72% 66.63%
1200 53.96% 67.91% 78.69% 65.67% 68.19% 62.70% 77.70% 69.79%
1300 56.84% 70.81% 81.25% 68.59% 71.08% 65.66% 80.34% 72.67%
1400 59.55% 73.46% 83.51% 71.26% 73.73% 68.36% 82.64% 75.24%
1500 62.09% 75.86% 85.49% 73.69% 76.13% 70.89% 84.70% 77.59%
1600 64.46% 78.03% 87.23% 75.90% 78.29% 73.20% 86.50% 79.74%
1700 66.69% 80.01% 88.74% 77.97% 80.27% 75.31% 88.08% 81.68%
1800 68.77% 81.81% 90.10% 79.83% 82.05% 77.23% 89.48% 83.42%
1900 70.72% 83.46% 91.29% 81.54% 83.67% 79.04% 90.71% 84.99%
2000 72.56% 84.95% 92.33% 83.11% 85.15% 80.68% 91.79% 86.43%
2100 74.28% 86.30% 93.26% 84.52% 86.49% 82.23% 92.77% 87.72%
2200 75.88% 87.54% 94.08% 85.85% 87.70% 83.62% 93.61% 88.89%
2300 77.39% 88.67% 94.79% 87.06% 88.81% 84.91% 94.37% 89.94%
2400 78.80% 89.69% 95.42% 88.17% 89.84% 86.11% 95.03% 90.89%
2500 80.13% 90.63% 95.97% 89.17% 90.76% 87.21% 95.61% 91.75%
2600 81.38% 91.49% 96.47% 90.10% 91.59% 88.22% 96.13% 92.54%
2700 82.54% 92.26% 96.90% 90.95% 92.35% 89.15% 96.59% 93.25%
2800 83.63% 92.96% 97.26% 91.71% 93.04% 90.00% 96.99% 93.88%
2900 84.65% 93.59% 97.59% 92.41% 93.67% 90.78% 97.34% 94.45%
3000 85.61% 94.16% 97.88% 93.06% 94.25% 91.51% 97.66% 94.98%
3100 86.51% 94.68% 98.13% 93.65% 94.78% 92.18% 97.93% 95.46%
3200 87.36% 95.16% 98.35% 94.20% 95.26% 92.80% 98.17% 95.90%
3300 88.15% 95.60% 98.55% 94.70% 95.69% 93.37% 98.39% 96.29%
3400 88.89% 96.00% 98.73% 95.15% 96.09% 93.89% 98.59% 96.65%
3500 89.58% 96.36% 98.88% 95.57% 96.43% 94.38% 98.75% 96.96%
3600 90.23% 96.69% 99.02% 95.95% 96.76% 94.83% 98.90% 97.25%
3700 90.84% 96.99% 99.14% 96.30% 97.05% 95.24% 99.03% 97.51%
3800 91.42% 97.26% 99.25% 96.61% 97.32% 95.61% 99.15% 97.75%
3900 91.96% 97.51% 99.34% 96.90% 97.57% 95.96% 99.25% 97.96%
4000 92.46% 97.73% 99.42% 97.17% 97.79% 96.28% 99.34% 98.15%
5000 96.04% 99.13% 99.84% 98.85% 99.14% 98.37% 99.81% 99.32%
6000 97.92% 99.65% 99.96% 99.53% 99.67% 99.28% 99.95% 99.75%
7000 98.92% 99.86% 99.99% 99.81% 99.88% 99.68% 99.99% 99.91%
8000 99.43% 99.94% 100.00% 99.92% 99.95% 99.86% 100.00% 99.97%
9000 99.70% 99.98% 100.00% 99.97% 99.98% 99.94% 100.00% 99.99%
10000 99.85% 99.99% 100.00% 99.99% 99.99% 99.97% 100.00% 100.00%
11000 99.92% 100.00% 100.00% 99.99% 100.00% 99.99% 100.00% 100.00%
12000 99.96% 100.00% 100.00% 100.00% 100.00% 100.00% 100.00% 100.00%
13000 99.98% 100.00% 100.00% 100.00% 100.00% 100.00% 100.00% 100.00%
14000 99.99% 100.00% 100.00% 100.00% 100.00% 100.00% 100.00% 100.00%
15000 99.99% 100.00% 100.00% 100.00% 100.00% 100.00% 100.00% 100.00%


Following is the average bankroll needed to hit a royal flush by game.

  • 9-6 Double Double Bonus: 1547.76
  • 9-6 Jacks or Better: 1056.41
  • Full pay Deuces Wild: 777.43
  • Not so ugly ducks Deuces Wild: 1123.74
  • 10-7 Double Bonus: 1049.43
  • 8-5 Bonus Poker: 1216.61
  • Full pay Joker Wild: 799.79
  • 8-5 Aces and Eights: 1002.47


Win of 200 or Bust



The following table shows the probability of achieving four deuces or better, which pays 200 for 1, in two Deuces Wild games. It also shows the same thing for achieving a five of a kind or better, which also pays 200, in full pay Joker Wild (kings or better).

Win of 200 or More or Bust

Bankroll FPDW NSUD FPJ
25 9.33% 7.83% 6.90%
50 17.78% 15.06% 13.30%
75 25.45% 21.72% 19.26%
100 32.39% 27.84% 24.82%
125 38.70% 33.50% 29.98%
150 44.43% 38.71% 34.80%
175 49.61% 43.51% 39.28%
200 54.31% 47.95% 43.45%
225 58.55% 52.03% 47.33%
250 62.42% 55.79% 50.95%
275 65.93% 59.25% 54.33%
300 69.11% 62.45% 57.46%
325 71.99% 65.39% 60.38%
350 74.60% 68.11% 63.11%
375 76.97% 70.61% 65.64%
400 79.12% 72.91% 68.00%
425 81.06% 75.04% 70.20%
450 82.83% 77.00% 72.24%
475 84.43% 78.80% 74.16%
500 85.88% 80.46% 75.93%
600 90.45% 85.90% 81.89%
700 93.55% 89.83% 86.39%
800 95.64% 92.66% 89.76%
900 97.06% 94.71% 92.30%
1000 98.01% 96.18% 94.22%
1100 98.65% 97.25% 95.66%
1200 99.09% 98.01% 96.73%
1300 99.38% 98.57% 97.55%
1400 99.58% 98.96% 98.15%
1500 99.72% 99.25% 98.61%
1600 99.81% 99.46% 98.96%
1700 99.87% 99.61% 99.22%
1800 99.91% 99.72% 99.41%
1900 99.94% 99.80% 99.56%
2000 99.96% 99.85% 99.67%
2100 99.97% 99.89% 99.75%
2200 99.98% 99.92% 99.81%
2300 99.99% 99.94% 99.86%
2400 99.99% 99.96% 99.89%
2500 99.99% 99.97% 99.92%
2600 100.00% 99.98% 99.94%
2700 100.00% 99.98% 99.96%
2800 100.00% 99.99% 99.97%
2900 100.00% 99.99% 99.98%
3000 100.00% 99.99% 99.98%


Following is the average bankroll needed to hit a win of 200 or more by game.

  • Full pay Deuces Wild: 255.90
  • Not so ugly ducks Deuces Wild: 306.79
  • Full pay Joker Wild: 351.40


Win of 25 or More or Bust



The following table shows the probability of a win of 25 or more for various bankroll sizes. I chose 25 because it the usual minimum pay for a four of a kind in games without wild cards. It is also a common win for a wild royal in deuces wild games.

For example, suppose a 9-6 Jacks or Better player puts a $100 bill into a 25¢ machine. What is the probability he will see a win of 25 bet units or more before going bust? $100 will buy 100/1.25 = 80 full bets on that machine. The table shows the probability of seeing a four of a kind win or more before the $100 is gone is 83.86%.

Win of 25 or More or Bust

Bankroll 9-6 DDB 9-6 JoB FPDW NSUD 10-7 DB 8-5 BP FPA8
10 10.36% 20.39% 16.00% 15.55% 12.25% 17.91% 17.92%
20 19.66% 36.61% 29.45% 28.69% 23.00% 32.61% 32.63%
30 27.99% 49.53% 40.74% 39.77% 32.44% 44.69% 44.70%
40 35.46% 59.82% 50.22% 49.14% 40.72% 54.59% 54.60%
50 42.15% 68.01% 58.18% 57.06% 47.99% 62.73% 62.73%
60 48.15% 74.53% 64.87% 63.73% 54.36% 69.41% 69.41%
70 53.53% 79.72% 70.50% 69.37% 59.96% 74.88% 74.89%
80 58.34% 83.86% 75.22% 74.14% 64.87% 79.38% 79.39%
90 62.67% 87.15% 79.18% 78.16% 69.18% 83.07% 83.08%
100 66.54% 89.77% 82.51% 81.55% 72.95% 86.11% 86.11%
110 70.01% 91.85% 85.31% 84.42% 76.26% 88.59% 88.60%
120 73.12% 93.51% 87.66% 86.84% 79.18% 90.64% 90.64%
130 75.90% 94.83% 89.63% 88.89% 81.73% 92.31% 92.32%
140 78.40% 95.89% 91.29% 90.62% 83.97% 93.69% 93.69%
150 80.64% 96.73% 92.69% 92.08% 85.93% 94.82% 94.82%
160 82.65% 97.39% 93.86% 93.31% 87.65% 95.74% 95.75%
170 84.45% 97.93% 94.84% 94.35% 89.17% 96.51% 96.51%
180 86.06% 98.35% 95.66% 95.23% 90.50% 97.13% 97.14%
190 87.50% 98.69% 96.36% 95.97% 91.66% 97.65% 97.65%
200 88.80% 98.95% 96.94% 96.59% 92.68% 98.07% 98.07%
250 93.52% 99.67% 98.72% 98.54% 96.19% 99.28% 99.28%
300 96.25% 99.89% 99.47% 99.37% 98.02% 99.73% 99.73%
350 97.83% 99.97% 99.78% 99.73% 98.97% 99.90% 99.90%
400 98.74% 99.99% 99.91% 99.88% 99.46% 99.96% 99.96%
450 99.27% 100.00% 99.96% 99.95% 99.72% 99.99% 99.99%
500 99.58% 100.00% 99.98% 99.98% 99.86% 99.99% 99.99%


Following is the average bankroll needed to hit a win of 25 or more by game.

  • 9-6 Double Double Bonus: 91.86
  • 9-6 Jacks or Better: 44.37
  • Full pay Deuces Wild: 57.85
  • Not so ugly ducks Deuces Wild: 59.66
  • 10-7 Double Bonus: 76.99
  • 8-5 Bonus Poker: 51.17
  • 8-5 Aces and Eights: 51.16


Methodology



A random simulation was used for this page. As usual, optimal player strategy is assumed.



Discussion of this topic can be found under the topic Bankroll Requirement to Hit a Royal at my forum at Wizard of Vegas.


Written by: Michael Shackleford

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