Which One is Worth More?

I’ve written about this subject a few times in the past, but not recently. If you’ve seen it before, check to see if you remember the correct answer. If you haven’t seen it before, now is the time to test yourself. All of us need to review from time to time to keep our skills sharp.

Assume you’re playing any version of Deuces Wild, where W represents a deuce, and we’re comparing the following two hands:  W 6♦ 7♦ 9♦ T♣ and W 6♦ 7♦ 9♦ 8♣.

In both hands you’re dealt a straight. In some versions of Deuces Wild, it’s correct to hold the straight flush draw and throw away the club. In other versions of the game, it’s better to hold all five cards. You should know which is correct in the Deuces Wild variations you play. If you don’t, use video poker software to look it up. This is not what I’m concerned about today.

I’m assuming you’re going to hold four cards to the straight flush, whether it’s the correct play or not. My question is:  Is there any difference in the value of W 6♦ 7♦ 9♦ when you throw away an off-suit T compared to when you throw away an off-suit 8? Take however long you need to make up your mind. I don’t mind waiting.

Before we give the answer to that problem, let’s look at the similar decision with K♠ 6♦ 7♦ 9♦ T♣ and K♠ 6♦ 7♦ 9♦ 8♣. Again, you’re going to hold the diamonds by themselves in both hands, right or wrong. (Never mind that in the hand with the off-suit 8, it’s typical to hold four cards to the straight.) My question is: Is there any difference in the value of 6♦ 7♦ 9♦ when you throw away an off-suit T (as well as the king) compared to when you throw away an off-suit 8 (again, as well as the king)? Take your time on this one too, if you need to.

And if your answers are not the same in the two examples, why not?

In the first case, when we are dealing with the 4-card straight flush (including the deuce as one of them), it doesn’t matter which card we throw away. If the club were a 3, for example, in those hands, there would be nine cards that would complete the straight — namely non-diamond 5s, 8s, and Ts. Since each hand has either an 8 or T in it, that will reduce the number of straights to eight in both cases. No other hand is affected differently.

In the second case, we’re drawing two cards, not one. Ignoring the deuces to start with, to reach a straight, we need a combination of 5-8 or 8-T, where at least one of them is a non-diamond. In the first example of the second case, K♠ 6♦ 7♦ 9♦ T♣, there are 15 ways to get 5-8. Any of the four 5s may be matched with any of the four 8s (making 16 combinations), but we have to subtract out the case where both the 5 and 8 are diamonds because that will yield a straight flush. To draw 8-T, there are only 11 ways. Any of the four 8s matched with any of the three Ts (because we were dealt the T♣ and it can’t come back after the draw), and subtracting out the one case where both the 8 and T are diamonds. So, we have a total of 26 ways to draw a natural straight.

In the second example of the second case, K♠ 6♦ 7♦ 9♦ 8♣, there are only going to be 11 ways to draw 5-8 and another 11 ways to draw 8-T. You need a natural 8, and there are only three of them still in the deck. So, this gives us 22 ways to draw a natural straight.

In both cases there will be 12 ways to draw a deuce and a non-diamond 5. There will be either eight ways to draw a deuce and a non-diamond 8 combined with 12 ways to draw a deuce and a non-diamond T, or 12 ways to draw a deuce and a non-diamond 8 combined with eight  ways to draw a deuce and a non-diamond T. Either way, that gives us 32 ways to draw a straight with exactly one wild card.

Adding them together, we have 58 ways to fill in the straight by holding the diamonds from K♠ 6♦ 7♦ 9♦ T♣ and only 54 ways to do it by holding the diamonds from K♠ 6♦ 7♦ 9♦ 8♣.

I strongly recommend you put these hands into video poker software and verify these numbers. Using the software is necessary when you’re trying to figure out how to attack various promotions, but you need to practice so you are able to figure these things out. Today’s exercise is a simple example that will give you some of this practice.