VIDEO POKER

A Look at Ten Cycles

This article has nothing to do with the recently completed Tour de France bicycle race.

A "cycle" in video poker is the number of times, on average, to complete a particular hand. For example, from Ah Kh Jh Qh 4c, you throw the four of clubs and have a 1-in-47 chance to complete the royal flush. The cycle on this hand is 47 hands. From 2h 2c 2d 7c 8c, you have a 2-in-47 chance to grab the fourth deuce. Drawing two cards to trips, then has a cycle of 23.5 hands.

Sometimes you see the term is modified as in "royal cycle" which means the number of starting hands to get a royal (40,390 hands for 9/6 Jacks or Better) or "quad cycle" (423 hands in 9/6 Jacks or Better), or maybe a "deuce cycle" (5,356 hands in NSU Deuces Wild).

If you talk about ten cycles, it would be 470 hands for 4-card royals, 403,900 hands for royals from scratch, 235 hands for filling in trips, etc. Largely it doesn't matter. If you're looking at the distribution of trips filled in in 235 hands, it's essentially identical to the number of Jacks or Better royals filled in in 403,900 hands.

In our first chart, we'll look at the distribution of filling in trips in 235 hands. In column B, we have 2/47 expressed in decimal form. Columns C represents the chances for exactly a certain number of quads. Column D represents the sum of what's in Column C

A	B	C	D	E
Number of Successes	Chance for 1 Hit	Chance for Total Hits	Cumulative Chances	Inverse of Total Hits
0	0.000025	0.0000	0.0000	22029.22
1	0.000025	0.0005	0.0005	2202.87
2	0.000025	0.0023	0.0028	440.56
3	0.000025	0.0076	0.0103	132.17
4	0.000025	0.0189	0.0293	52.87
5	0.000025	0.0378	0.0671	26.43
6	0.000025	0.0631	0.1301	15.86
7	0.000025	0.0901	0.2202	11.10
8	0.000025	0.1126	0.3328	8.88
9	0.000025	0.1251	0.4579	7.99
10	0.000025	0.1251	0.5830	7.99
11	0.000025	0.1137	0.6968	8.79
12	0.000025	0.0948	0.7916	10.55
13	0.000025	0.0729	0.8645	13.72
14	0.000025	0.0521	0.9165	19.20
15	0.000025	0.0347	0.9513	28.80
16	0.000025	0.0217	0.9730	46.09
17	0.000025	0.0128	0.9857	78.35
18	0.000025	0.0071	0.9928	141.03
19	0.000025	0.0037	0.9965	267.96
20	0.000025	0.0019	0.9984	535.94

In our second chart, we will look at the chances of filling in royals in 400,000 hands, given the chance to complete one royal is 1-in-40,000

A	B	C	D	E
Number of Successes	Chance for 1 Hit	Chance for Total Hits	Cumulative Chances	Inverse of Total Hits
0	0.000025	0.0000	0.0000	22029.22
1	0.000025	0.0005	0.0005	2202.87
2	0.000025	0.0023	0.0028	440.56
3	0.000025	0.0076	0.0103	132.17
4	0.000025	0.0189	0.0293	52.87
5	0.000025	0.0378	0.0671	26.43
6	0.000025	0.0631	0.1301	15.86
7	0.000025	0.0901	0.2202	11.10
8	0.000025	0.1126	0.3328	8.88
9	0.000025	0.1251	0.4579	7.99
10	0.000025	0.1251	0.5830	7.99
11	0.000025	0.1137	0.6968	8.79
12	0.000025	0.0948	0.7916	10.55
13	0.000025	0.0729	0.8645	13.72
14	0.000025	0.0521	0.9165	19.20
15	0.000025	0.0347	0.9513	28.80
16	0.000025	0.0217	0.9730	46.09
17	0.000025	0.0128	0.9857	78.35
18	0.000025	0.0071	0.9928	141.03
19	0.000025	0.0037	0.9965	267.96
20	0.000025	0.0019	0.9984	535.94

The only column that is markedly different is B. In our first chart we see 0.0426, which is the same as 1/23.5, rounded to four decimal places. In the second chart we see 0.000025, which is 1/40,000. Otherwise the charts are similar.

The chance of receiving 6 quads in 10 cycles is 6.17%. The chance of receiving 6 royals in 10 cycles is 6.31%. Not identical, but fairly close. The chances of receiving 12 hits respectively are 9.64% and 9.48%. Again, fairly close.

Both charts were created using the BINOMDIST function in an Excel spreadsheet. Creating another chart for the 1-in-423 chances of getting a quad would be quite simple and yield similar results.

The chart shows way too many significant digits to be useful. Let's eliminate some and we get the following:

A	C	D
Number of Successes	Chance for Total Hits	Cumulative Chances
0	0.00	0.00
1	0.00	0.00
2	0.00	0.00
3	0.01	0.01
4	0.02	0.03
5	0.04	0.07
6	0.06	0.13
7	0.09	0.22
8	0.11	0.33
9	0.13	0.46
10	0.13	0.58
11	0.11	0.70
12	0.09	0.79
13	0.07	0.86
14	0.05	0.92
15	0.03	0.95
16	0.02	0.97
17	0.01	0.99
18	0.01	0.99
19	0.00	1.00
20	0.00	1.00

Although this was created from the second chart, we've already seen that the numbers would be essentially identical if we started from the first chart.

Many players believe that if you play 10 royal cycles, it means you SHOULD get exactly 10 royals, and if you get fewer than that number it means that someone (possibly the casino, possibly Bob Dancer) is gypping you somehow. Looking at the chart, you can see that there is only a 13% chance of getting exactly 10 royals, and there is a 46% chance that you'll get fewer than 10 royals.

Also, note that these charts relate to the NEXT 10 cycles, not the LAST 10 cycles. In the last 10 cycles, you have had a specific number of hits --- perhaps 4, perhaps 13, whatever. It is just one number. There is no distribution of hits.

But in the next 10 cycles, these are the appropriate numbers --- assuming you are playing the appropriate strategy.