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Bob Dancer writes a video poker column for beginners to experts. He also writes a column with Jeffrey Compton, "Player's Edge", featuring information on promotions at various Las Vegas casinos. Player's Edge is published each Friday in the Neon section of the Las Vegas Review-Journal. Click here to send Bob Dancer an e-mail.

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June 02, 2009

Some Math Behind Quick Quads

In last week's column I stated without proof that from 'A235' 3 we hold A233, but from 'A235' 5 we hold 'A235'. Let's look at the math behind these statements. The math is not difficult and solid video poker players need to have a certain facility with this sort of calculation.

From 'A235' 3, the following chart shows the possibilities:

'A235'			How Many
	Hand	Payoff	Times?	Product	EV
	SF	250	1	250
	FL	30	8	240
	ST	20	3	60
	High Pair	5	3	15
	No Pay	0	32	0
				565
					12.02128
332A
	QQ	400	2	800
	2P	5	6	30
	No Pay	0	39	0
				830
					17.65957

The top part of the chart shows the possibilities when you draw one card to 'A235'. You get a straight flush once, a regular flush eight times, a straight three times, and a high pair three times. Any other draw gives us a score of zero. I strongly suggest that you verify these counts for yourself. We multiply the number of occurrences by the pay schedule values and sum them up and come up with a total 565. Since there are 47 possible draws, the average is 12.02128. (This is sometimes called EV for expected value. Technically EV should be calculated on a 1-coin basis. Here it is calculated on a 6-coin basis.)

The bottom part of the chart shows the results when we draw to the "Quick Trip" 332A. We score 400-coin Quick Quads twice, two pair six times, and otherwise score nothing. This adds up to 830 and gives us an EV of 17.65957. Since this is much higher than 12.02128, we should hold 332A.

Another hand that gives us the exact same choice 'A234' 3. Since four 4s pay the same as four 3s, the two hands 'A345' 4 and 'A234' 4 also are played the same way for the same mathematical reason.

From 'A235' 5, holding the 4-card straight flush gives us the same numbers as before, again yielding an EV of 12.02128. Holding the 5532 Quick Trip gives us the following:

5532
	QQ	260	2	520
	2P	5	6	30
	No Pay	0	39	0
				550
					11.70213

The difference is that the Quick Quads now only return 260 coins instead of 400. The EV of 11.70213 is lower than the 12.02128 we get from holding the 4-card straight flush, so this time the straight flush is the correct play.

The two hands 'A245' 5 and 'A345' 5 are played the same way for the same reason.

And interesting choice occurs with the hand '2346' 6. Compare the following chart with the first one:

'2346'			How Many
	Hand	Payoff	Times?	Product	EV
	SF	250	1	250
	FL	30	8	240
	ST	20	3	60
	No Pay	0	35	0

				550
					11.70213
6642
	QQ	260	2	520
	2P	5	6	30
	No Pay	0	39	0
				550
					11.70213

The difference between them is that now we have no chance to get a high pair, so that reduces the EV to 11.70213. Coincidentally the EV of the 6642 Quick Trip is also 11.70213. It's relatively rare that we get an exact tie in video poker, but such is the case here.

Other hands that are also exhibit a tie between a 260-coin Quick Quad and a 4-card inside straight flush with no high cards are: '2356' 6, '3457' 7, and '3467' 7.

I will leave it to the reader as an exercise, but when you have an open-ended 4-card straight flush along with a Quick Trip (namely '2345' 5), going for the 4-card straight flush is correct by a large margin. After this article it shouldn't be difficult to figure out. Good players make the effort to do these things.