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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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Jan. 29, 2008

The Value of Three Deuces in Wild Royals

In all forms of Deuces Wild, there are ten 3-deuce wild royals, namely WWW 'AK', WWW 'AQ', WWW 'AJ', WWW 'AT', WWW 'KQ', WWW 'KJ', WWW 'QJ', WWW 'QT', and WWW 'JT'. I'm using my standard notation of the 'W' (for wild card) to stand for the deuce and cards within quotation marks being suited with each other. In some of my classes I ask my students which of these has more value. Although many miss this question the first time they hear it, the answer is very simple. They all have exactly the same value --- which is usually 125 or 100 coins.

There are versions of Deuces Wild that pay premiums for four deuces. Instead of the standard 200-for-1, Double Deuces pays 400-for-1, Loose Deuces returns 500-for-1, and Triple Deuces pays 600-for-1. In some of these games, depending on the exact pay schedule, in some or all of the wild royals listed above, the correct play is just the three deuces.

I found myself looking at this hand as I was preparing the Double Deuces class I'll be teaching February 6 in my new semester at the El Cortez. Although the correct play in the 99.62% version of Double Deuces I'll be teaching at the El Cortez is to hold all five cards of the wild royal in all ten cases listed above, I noticed that the value of three deuces differed --- but not how I expected it to.

Before reading further, go ahead and rank the value of these hands. That is, if you hold WWW from WWW 'AQ', is the value of WWW higher or lower than if the original hand was WWW 'AK'? Although the ranking turns out to be rather simple to explain, my guess is that not one player in 1,000 can get the ranking correct before reading further.

When you draw two cards to WWW, you are guaranteed to receive at least 4-of-a-kind, although you might receive a straight flush, 5-of-a-kind, wild royal flush, or four deuces. Of the 1,081 possible draws, you're going to receive each of these end results the following number of times.

	4K	SF	5K	WRF	4D

WWW 'AK'	816	120	66	43	46
WWW 'AQ'	817	119	66	43	46
WWW 'AJ'	818	118	66	43	46
WWW 'AT'	819	117	66	43	46

WWW 'KQ'	815	121	66	43	46
WWW 'KJ'	816	120	66	43	46
WWW 'KT'	817	119	66	43	46

WWW 'QJ'	817	119	66	43	46
WWW 'QT'	818	118	66	43	46

WWW 'JT'	819	117	66	43	46

There is no variation at all in the number of times you receive 5-of-a-kind, wild royal flush, or four deuces from each of the ten starting hands. The number of times you end up with a straight flush, however, varies. The number of times you get 4-of-a-kind changes correspondingly with the number of times you get a straight flush --- since all of the categories together must add up to 1,081.

I then set out to determine WHY the number of straight flushes varied. It turned out to be a rather simple explanation.

If we could draw from a pack of 48 cards (i.e. all cards remaining in the deck except the four deuces), it would turn out that there were exactly 124 straight flushes in all of these cases. But we are really looking at drawing from 46 cards here. (We normally consider 47 cards --- the 52 original minus the five cards we are looking at --- but here we are also excluding the fourth deuce from consideration as that affects all of these combinations equally.) Hence we need to look at how each of the cards A, K, Q, J, T individually reduce the number of possible straight flushes from the potential of 124.

When you are dealt an ace and throw it away, the straight flushes you lose out on are WWW 'A3', WWW 'A4', and WWW 'A5'.

When you are dealt a king and throw it away, the only straight flush you lose out on is WWW 'K9'.

When you are dealt a queen and throw it away, you lose out on WWW 'Q9' and WWW 'Q8'.

When you are dealt a jack and throw it away, you lose out on WWW 'J9', WWW 'J8', and WWW 'J7'.

Finally, when you are dealt a ten and throw it away, you lose out on WWW 'T9', WWW 'T8', WWW 'T7', and WWW 'T6'.

It turns out that throwing away a ten is the most costly. Perhaps surprisingly, discarding the jack and ace are tied for second place. Next comes discarding the queen, and discarding a king is the least costly.

Adding these together we find each of the combinations miss out on the following straight flushes:

	SF
WWW 'AK'	120	'A3'	'A4'	'A5'	'K9'
WWW 'AQ'	119	'A3'	'A4'	'A5'	'Q9'	'Q8'
WWW 'AJ'	118	'A3'	'A4'	'A5'	'J9'	'J8'	'J7'
WWW 'AT'	117	'A3'	'A4'	'A5'	'T9'	'T8'	'T7'	'T6'

WWW 'KQ'	121	'K9'	'Q9'	'Q8'
WWW 'KJ'	120	'K9'	'J9'	'J8'	'J7'
WWW 'KT'	119	'K9'	'T9'	'T8'	'T7'	'T6'

WWW 'QJ'	119	'Q9'	'Q8'	'J9'	'J8'	'J7'
WWW 'QT'	118	'Q9'	'Q8'	'T9'	'T8'	'T7'	'T6'

WWW 'JT'	117	'J9'	'J8'	'J7'	'T9'	'T8'	'T7'	'T6'

The highest valued three deuces are the ones that miss out on the fewest straight flushes. Another way to say that is that the ones that end up with more straight flushes are more valuable than those that end up with less. So the ranking of hands, assuming we are just going to hold the three deuces, is:

First:	WWW 'KQ'

Tied for second:	WWW 'AK', WWW 'KJ'

Tied for fourth:	WWW 'AQ', WWW 'KT', WWW 'QJ'

Tied for seventh:	WWW 'AJ', WWW 'QT'

Tied for ninth:	WWW 'AT', WWW 'JT'

Who would have thunk?