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Provided by The Wizard of Odds

Video Poker Appendix 1

This appendix addresses the question of bankroll size Vs. risk of ruin in video poker. For those who don't know, the risk of ruin is the probability of losing an entire bankroll. The following tables show the number of betting units required according to the desired risk of ruin, the game, and cash back. A "betting unit" is five coins, for example a betting unit would be $1.25 for a 25 cent machine player.

As an example the full play deuces wild player, with 0.25% cash back, would need a bankroll of 3345 units to have a probability of ruin of 5%. See the following chart to find this number. These numbers may seem high compared to other sources based on ruin before some other event is achieved. The tables below are for ruin at any time over an infinite period of time and thus have no successful terminating event, other than reaching an infinite bankroll. Consequently these tables are best used by the player considering establishing a bankroll for an indefinite period of play

Deuces Wild

The following table applies to "full pay" deuces wild. This pay table can be found in my video poker tables but is generally marked by paying 5 for a four of a kind. The expected return on this game is 100.76% and the standard deviation is 5.08.

Risk
of Ruin		 Cash Back
0% 0.25% 0.50% 0.75% 1.00%
50.00% 1061 774 601 486 404
40.00% 1402 1023 795 643 534
30.00% 1843 1344 1044 844 702
20.00% 2463 1797 1396 1129 939
10.00% 3524 2571 1997 1615 1343
7.50% 3964 2892 2247 1817 1511
5.00% 4585 3345 2598 2101 1747
2.50% 5646 4119 3200 2587 2151
1.00% 7048 5142 3994 3230 2686
0.50% 8109 5916 4596 3716 3090
0.25% 9170 6690 5197 4202 3494
0.10% 10572 7713 5992 4845 4029
0.05% 11633 8487 6593 5331 4433
0.025% 12694 9261 7194 5817 4837
0.01% 14096 10284 7989 6460 5372

Double Bonus

The following table applies to "10/7" double bonus. This pay table can be found in my video poker tables but is generally marked by paying 7 for a flush and 10 for a full house. The expected return on this game is 100.17% and the standard deviation is 5.32.

Risk
of Ruin		 Cash Back
0% 0.25% 0.50% 0.75% 1.00%
50.00% 5761 2254 1391 999 776
40.00% 7615 2980 1839 1321 1026
30.00% 10006 3916 2417 1736 1348
20.00% 13376 5235 3230 2320 1802
10.00% 19137 7489 4622 3320 2578
7.50% 21528 8425 5199 3735 2900
5.00% 24897 9744 6013 4319 3354
2.50% 30658 11998 7404 5319 4130
1.00% 38273 14978 9244 6640 5155
0.50% 44034 17233 10635 7639 5931
0.25% 49795 19487 12026 8639 6707
0.10% 57410 22467 13865 9960 7733
0.05% 63171 24722 15257 10959 8509
0.025% 68931 26976 16648 11958 9285
0.01% 76547 29956 18487 13280 10311

Jacks or Better

The following table applies to "full pay" jacks or better. This pay table can be found in my video poker tables but is generally marked by paying 6 for a flush and 9 for a full house. The expected return on this game is 99.54% and the standard deviation is 4.42.

Risk
of Ruin		 Cash Back
0.50% 0.75% 1.00% 1.25% 1.50%
50.00% 15454 2198 1130 735 530
40.00% 20429 2906 1493 971 701
30.00% 26843 3819 1962 1276 921
20.00% 35883 5105 2623 1706 1231
10.00% 51337 7303 3752 2441 1761
7.50% 57751 8216 4221 2746 1982
5.00% 66791 9502 4882 3176 2292
2.50% 82245 11700 6011 3911 2822
1.00% 102674 14606 7505 4882 3523
0.50% 118128 16805 8634 5617 4053
0.25% 133581 19003 9764 6352 4584
0.10% 154010 21910 11257 7323 5284
0.05% 169464 24108 12386 8058 5815
0.025% 184918 26306 13516 8793 6345
0.01% 205347 29213 15009 9764 7046

Methodology

An entirely mathematical approach was used to create the above tables. The theory was similar to that of the solution of problem 72 in my site of math problems. Briefly if p is the probability of ruin with 1 unit then p2 is the probability of ruin with 2 units, p3 is the probability of ruin with 3 units, and so on. With the known probabilities for the outcome of each hand an equation could be set up to solve: p=sum over all possible outcomes of pri * pri, where pri is the probability of hand i and ri is the return for hand i. Using an iterative process I solved for p.

The cash back had to be factored using a little finesse. In all cases I assumed the player redeemed his cash back whenever he received a particular hand, for example five of a kind in deuces wild. It was also assumed he played the expected number of hands between such cash-back hands every time. In the case of deuces wild 312.34 hands are played on average for each five of a kind.

Other Sources

For the player interested in the risk of ruin before achieving a desired goal, either receiving one royal flush or playing a specified number of hands, I would recommend Video Poker Optimum Play by Dan Paymar.

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