VIDEO POKER

Power of the Pack in Full Pay Deuces Wild

Letting a W stand for a wild card, why would you hold, in full pay Deuces Wild (i.e. where 4-of-a-kinds return 5 and 5-of-a-kinds return 15), WAK from W Ah Kh 7s 8d but not from W Ah Kh 4s 3d? After all, the value of the 3-card royal flush hasn't changed between the two hands. Most of the value from drawing two cards to WAK comes from drawing other aces, kings, deuces, cards of the same suit (in this case hearts) to get a flush or wild royal, and cards equaling a ten or higher to get a straight. Since we are discarding an off-suit 78 in the first case and off-suit 34 in the second, none of the discards affect the value of the WAK. Indeed, in terms of 5-coin expected value, the value of WAK in the first hand is 5.2313 coins and the value of WAK in the second hand is also 5.2313 coins.

It certainly isn't obvious how a combination worth 5.23 coins can be worth holding in one case and not in the other. At least not until we look at what else is in the hand. The other choice in the hand is to hold the W by itself. It turns out that the sole W is worth more than 5.23 coins in one case and less than 5.23 coins in the other. Once you know this, it's easy to understand why you hold WAK in one case but not the other. But the question still remains WHY is the deuce worth more in one case than the other.

The answer lies in a concept I like to call the "Power of the Pack." In Liam W. Daily's and my Winner's Guide series, we use the term "deck" to refer to the 52 initial cards, and the term "pack" to refer to the 47 cards remaining after the first five cards are dealt. When we draw four cards to a single deuce, everything other than a natural royal flush is possible. There are 178,365 different combinations we can draw --- some of them good and some of them worthless. The difference between when we discard the 78 compared to when we discard the 34 is due to the number of the straights and straight flushes involved. First let's look at the numbers, and then I'll explain them.

		W Ah Kh 7s 8d		W Ah Kd 3s 4d
	Value of WAK	5.2313 coins		5.2313 coins
	Value of W	5.1772 coins		5.2788 coins
	Number of Straights	13,554		15,485
	Number of Straight Flushes	806		891

There is a difference of over a tenth of a coin between the value of drawing to the single deuce. Assuming you are playing for quarters, that means 2.5¢. Is that enough to worry about? It is for me.

So why are the number of straights different? Consider, for example, the straight JT987, where one or two of the cards are deuces. Holding just the deuce from W Ah Kh 7s 8d, there are four Js, four Ts, four 9s, three 8s, three 7s, and four deuces still in the deck. To get this straight, you need to draw an acceptable four cards out of the 22 out there (4+4+4+3+3+4 = 22). To get this straight holding just the deuce from W Ah Kh 3s 4d, there are 24 cards to work with because the 7s and the 8d are still in the deck.

There are some straights that the 34 poisons more than the 78 does, but not many. It is harder to get the QJT98, JT987, T9876, 98765, 87654 when you discard the 7 and the 8, and easier to get the 76543 and 543WA straight. Getting the AKQJT and KQJT9 straights are equally likely in these two cases.

In addition to these hands, there are other types of hands in full pay Deuces Wild where the power of the pack comes into play. Such as holding KQ from Kh Qh 9s 8d 7c and drawing five new cards from Kh Qh 9s 4d 3c, or holding KQ from Kd Qd 7s 5h 3c but drawing five new cards from Kd Qd 5s 4h 3c.

In NSU, there are a number of cases where the Power of the Pack is important. One of these are holding W45 from W 4h 5h 6s Qs but drawing to the sole W from W 4h 5h 3s Qs. Another is holding just the W from W 5h 6d 7s Ah, but holding W567 from W 5h 6d 7s Jh. A third case is holding KQ from Kh Qh Js 7s 5h but drawing five cards to Kh Qh Js 4s 3h. There are other cases as well in this game.

In Kings or Better Joker Wild, one example of a distinction that depends on the Power of the Pack is holding QT from Qc Tc Jh 6s 3c but drawing five cards from Qc Tc Jh 4s 3c. Holding mid-cards (5-T) with a joker in this game is largely a Power of the Pack discussion.