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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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Mar 20, 2007

How Do You Figure?

I was playing NSU Deuces Wild recently, sitting near a fellow professional. We were both waiting to be paid for a set of four deuces. He had started with three deuces and I'd started with two. He asked me whether you got four deuces more frequently starting from two deuces or three. Obviously it's easier to get four deuces once you already have three than it is when you start with two, but you draw to two deuces a lot more times. I didn't know how it all worked out, but it sounded like an interesting problem.

Obviously the correct answer depends on what version of Deuces Wild you are playing. In Full Pay Deuces Wild, for example, letting a W stand for a deuce, you toss the threes from WWW33 (which gives you a chance to hit four deuces) whereas from the same hand in NSU Deuces Wild you hold all five cards (which doesn't give you a chance to connect on four deuces this hand.)

To start with, let's look at a breakdown of starting hands in Deuces Wild.

	No Deuces	1 Deuce	2 Deuces	3 Deuces	4 Deuces
No Score	1369920	105840	0	0	0	1475760
Queens or Better	190080	358848	0	0	0	548928
Two Pair	95040	0	0	0	0	95040
3-of-a-kind	42240	253440	59400	0	0	355080
Straight	8160	34272	19800	0	0	62232
Flush	3136	7376	3960	0	0	14472
Full House	3168	9504	0	0	0	12672
4-of-a-kind	528	8448	19008	3568	0	31552
Straight Flush	28	464	1080	496	0	2068
5-of-a-kind	0	48	288	288	0	624
Wild Royal Flush	0	80	240	160	0	480
Four Deuces	0	0	0	0	48	48
Natural Royal Flush	4	0	0	0	0	4

	1712304	778320	103776	4512	48	2598960

I devised this chart back when I was figuring out returns for Double Pay video poker, which pays you on the deal as well as paying you on the draw. That's the reason you see the "Queens or Better" category, which is a paying category in that game but not in regular Deuces Wild. It doesn't affect what we're doing here because you only get that hand in the "no deuces" or "one deuce" column, and we're looking at the next two columns here.

Let's look at the 3-Deuce column first. Our chart tells us that we are dealt three deuces 4,512 times, and it also gives us the number of times we'll 4-of-a-kind, straight flush, and other hands. We'll draw to every hand where it shows 4-of-a-kind (which means a hand like WWW3K). In FPDW we'll also draw to the 5-of-a-kinds in the WWW33-WWW99 range, which happen 168 times. This means in FPDW, we draw on 3,568 + 168 = 3,736 hands out of the 2,598,960 possible starting hands. We'll connect two times out of 47 (that is, we can draw the fourth deuce in the first empty hole or the second). Multiplying this out, we get (3,736 * 2) / (2,598,960 * 47) = 6.117 e-5. (The e-5 at the end refers to the number of zeroes that should be in front of the 6.117. This is the same as 0.00006117, or 0.006117%, which is the same as one time in 16,347). In NSU, since we hold all five of a kinds, this number is (3,568 * 2) / (2,598,960 * 47) = 5.842 e-5, which is the same as one time in 28,026.

Now we look at two deuces. Of the 103,776 times we draw two deuces (about once every 25 hands); in both games we're going to hold all 4-of-a-kinds, straight flushes, 5-of-a-kinds, and wild royal flushes. This subtracts out 20,616 hands and leaves us 83,160 hands we need to examine.

In both games we also hold 4-card royal flushes, like WW'AK' or WW'QJ', where the quote marks means the cards are suited with each other. First, let's count the number of WW'AK' hands. We don't include the suited Q, J, or T because that would give us a wild royal, which we've already counted. We don't include an A or K of another suit, because that gives us a 4-of-a-kind, which we've also already counted. With suited cards, we have 8 of them (3,4,5,6,7,8,9). With unsuited cards, we count 10 of them (3,4,5,6,7,8,9,T,J,Q), which can be any of three different suits. Therefore, there are 38 different cards that we can have with a specific WW'AK' that will give us a combination that we'll hold.

There are 6 ways to have two deuces (suitwise: cd, ch, cs, dh, ds, hs), and 4 different suits to have the 'AK'. There are also four different A-high 2-card royals ('AK', 'AQ', 'AJ', and 'AT') so multiplying these all out gives us 6 * 4 * 4 * 38 = 3,648 two-deuce ace-high 4-card royals.

There are only three different K-high 4-card royals (WW'KQ', WW'KJ', and WW'KT'). The number of off-suit cards we can have doesn't change, but with the suited cards, a 9 will give us a straight flush and that has already been excluded. This gives us 6 * 4 * 3 * 37 = 2,664.

For Q-high (WW'QJ' and WW'QT'), we also exclude the suited 8, so we have 6 * 4 * 2 * 36 = 1,728. For J-high, we also exclude the suited 7, so we have 6 * 4 * 1 * 35 = 840. Adding up all of the 4-card royals that we hold we have 3,648 + 2664 + 1,728 + 840 = 8,880.

Both games also hold certain 4-card straight flushes. There are four different ones with no insides, specifically WW'67', WW'78', WW'89', and WW'9T'. These all have the same value, so we'll count the cases for any one of them (say WW'67') and multiply by four so that we cover all of them. For cards suited with the '67', we'll exclude six of them that make a straight flush (specifically the 3,4,5,8,9, and T) which leaves four that we can be dealt where we will draw to the straight flush (namely the J, Q, K, and A). For unsuited cards, as before we have ten of them (3,4,5,8,9,T,J,Q,K, and A) of three different suits. That gives us a total of 34 cards. There remain six different ways to start with two deuces and four different suits for the straight flush cards, and four types of these straight flushes, so 6 * 4 * 4 * 34 = 3,264.

In NSU we also hold 4-card straight flushes with one inside, specifically WW'45', WW'56', WW'57', WW'68', WW'79', WW'8T', and WW'9J'. All seven of these have the same value, so we need to look at any one of them (say WW'79') and multiply by seven. With suited cards, there are five of them that yield the straight flush (namely 5, 6, 8, T, and J) which leave five that give us a hand to draw to (namely 3, 4, Q, K, and A). There remain ten unsuited ranks, times three suits, that we draw to, totaling 35 different cards. This gives us 6 * 4 * 7 * 35 = 5,880.

So for FPDW, we start with the 83,160 hands and subtract out the hands for 4-card wild royals (8,880) and 4-card straight flushes with no insides (5,880) which total up to 83,160 - 8,880 - 3,264 = 71,016 different ways to draw to two blank deuces. Since each time we draw three cards to two deuces there are 3/1,081 times we convert, this totals out to be (71,016 * 3) / (2,588,960 * 1,081) = 7.583 e-5 (or once every 13,187 hands). This is more frequent than the once every 16,347 hands we convert from three deuces in this game.

In NSU, we need to subtract the 4-card straight flushes with one inside from the 71,016 hands, which gives us 71,016 - 5,880 = 65,136. The total equation becomes (65,136 * 3) / (2,598,960 * 1,081) = 6.955 e-5 (or once every 14,377 hands). This is almost twice as frequent as the once in every 28,026 hands we end up with four deuces starting with three deuces.

So in both games, it is more likely that you got your four deuces starting with two rather than three. Although this answer is somewhat interesting, the real purpose of the article is to show you how this is calculated.