VIDEO POKER

A Question from Double Double Joker Poker

I have never written about this game before, and have just recently started to look at it. It is dealt from a 54-card deck that contains two jokers. Although difficult, the 9/5 version returns just a tick under 100% with a reasonably low variance.

Letting a W (for wild card) stand for a joker, let me ask you about three similar hands:

a. W W Ah Ks 6c
b. W W Ah Qs 6c
c. W W Ah Js 6c

It won't surprise you to know that you always hold every joker in this game. And please accept that with two jokers, holding a single honor card is an eligible play --- meaning that you'll hold an honor card with the two jokers before holding the two jokers by themselves, but on a hand such as W W Ks 5d 5c, WW55 has a higher value than WWK. Therefore, on the first hand the choice is WWA or WWK, on the second it is WWA or WWQ, and on the third it is WWA or WWJ. Place your bets. I'll give you a hint. The correct answer is not obvious.

In all three hands, notice that the 6c was carefully chosen so as to provide no straight or flush interference with any of the other cards. It could have equally been the 6d, but no other card in the deck would suffice. If the card were higher than the 6, it would affect the straight opportunities of the J, at least. If the card were lower than the 6, it would affect the straight opportunities of the A. And any heart or spade would affect flush or straight flush opportunities.

Since there are 12 hearts and 12 spades left in the 49-card pack (i.e, the 54-card deck minus the 5 cards we are looking at), the sum of (wild royal flushes + straight flushes + flushes) remains constant --- and that number is 66. (I calculate it by 12 * 11 / 2 = 66.)

In each of the draws, there are six potential wild royal flushes. When we draw to the W W Ah, for example, we could draw Kh Qh, Kh Jh, Kh Th, Qh Jh, Qh Th, or Jh Th and end up with the wild royal every time . You may easily verify that there are six ways for a wild royal starting from WWK, WWQ, and WWJ.

In the matter of straight flushes, however, the number of the combinations is different, but not in an easily predictable way. In non-wild-card games, a J has more potential straight flushes than a Q, which in turn has more potential straight flushes than a K, which has about the same number of potential straight flushes as an A because of A2345. Adding wild cards, however, changes the pattern.

Assuming the correct suits, from WWA we can draw 23, 24, 25, 34, 35, and 45 = 6 different straight flushes. To WWK, we can draw Q9, J9, T9 = 3 different straight flushes. To WWQ, we can draw K9, J9, J8, T9, T8, 98 = 6 different straight flushes, and to WWJ we can draw K9, Q9, Q8, T9, T8, T7, 98, 97, 87 = 9 different straight flushes.

Straights will follow the same ranks as either the wild royals or the straight flushes, except that at least one of the cards will not be suited with the others. Therefore the combinations yielding the most straight flush draws will also yield the most straight draws. The number of 3-of-a-kinds, 4-of-a-kinds, and 5-of-a-kinds remains the same on each of the draws.

So the answers to our quiz above are:

a. WWA
b. WWA = WWQ
c. WWJ

Notice there was one tie --- but that was only in the unusual case where the fifth card provided neither flush nor straight interference. Either flush or straight interference is sufficient to tilt the correct answer, with flush interference being more important than straight interference. That is,

From W W Ah Qs 8s, choose WWA
From W W Ah Qs 7d, choose either
From W W Ah Qs 4d, choose WWQ
From W W Ah Qs 4s, choose WWA
From W W Ah Qs 9h, choose WWQ

Finally, although it's not part of today's lesson, one type of hand must be included because it would be easy to misplay,

From W W Ah Qs 5h, choose WWA5