I periodically check the wizardofodds.com strategy calculator to make sure I’ve gotten every one of the special cases mastered. The format used in that application isn’t as user friendly (to me) as I’d like, although I’m sure that for some players it is easy to understand.
For example, in discussing the hands where there is a suited AQ and an unsuited AQJT inside straight draw, he lists the following as all the cases where you go for the straight (although the site has it in one column not two, and the labels in front of each hand were added here by me so I could refer to them more clearly):
A 2♣10♦J♦Q♣A♣ H 5♣10♦J♥Q♣A♣
B 2♣10♦J♥Q♣A♣ I 6♣10♦J♦Q♣A♣
C 3♣10♦J♦Q♣A♣ J 6♣10♦J♥Q♣A♣
D 3♣10♦J♥Q♣A♣ K 7♣10♦J♦Q♣A♣
E 4♣10♦J♦Q♣A♣ L 7♣10♦J♥Q♣A♣
F 4♣10♦J♥Q♣A♣ M 8♣10♦J♦Q♣A♣
G 5♣10♦J♦Q♣A♣ N 8♣10♦J♥Q♣A♣
In Example A, he’s not referring to clubs and diamonds specifically. He’s including all cases where the ‘AQ2’ are suited with each other and the ‘JT’ are also suited with each other. There are twelve hands that are included in his shorthand — because the ‘AQ2’ can be any of the four suits and once that suit is chosen, there are three choices for the other suit.
In Example B, there are twenty-four hands included, because here the jack and ten aren’t suited with each other, so there are four suits for the ‘AQ2’, three suits for the jack and two suits for the ten. Multiplying 4*3*2 gives twenty-four possibilities out of the 2.6 million starting hands.
What all of these hands have in common is that the fifth card is suited with the suited ‘AQ’ — what I call a flush penalty. Whether the jack and ten are suited or not isn’t relevant to me because a suited ‘JT’ is lower on my strategy list than both a suited ‘AQ’ and an inside straight with three high cards.
On my chart I indicate this as ‘AQ’ with fp ‹ ST4 3h1i. Since ‘AK’, ‘AQ’, and ‘AJ’ all have the same value, I lump them together as ‘AH’ (where the H stands for a high card lower than an ace — and a high card is defined as one where a pair of them gives your money back). My chart says ‘AH’ with fp ‹ ST4 3h1i — where the Wizard of Odds (WoO) site gives you three lists — one for ‘AK’, one for ‘AQ’, and one for ‘AJ’. If the WoO way helps you understand, that’s fine with me. Everybody’s brain works in a slightly different way.
I actually went to the WoO site to look at the hand ’79T’ with an off-suit ace — when to hold the ace by itself and when to hold ’79T’. As many times as I’ve looked at the WoO site and tried to make my list perfect, I just learned a new exception.
Look at the following (and the ‘identical’ list can be found with ’78T’ versus an ace). These are the only hands where you hold the ’79T’ rather than the ace:
O 2♣7♦9♦10♦A♣
P 3♣7♦9♦10♦A♣
Q 3♣7♦9♦10♦A♥
R 4♣7♦9♦10♦A♣
S 4♣7♦9♦10♦A♥
T 5♣7♦9♦10♦A♣
U 7♣9♣10♣Q♦A♥
V 7♣9♣10♣K♦A♥
Hands U and V have to do with straight penalties to the ace, and you don’t see any hands with a 6 or jack because those involve straight penalties to the ’79T’. I want to concentrate on the other six hands.
Notice that when there is an ace straight flush penalty (O, P, R, and T), you always hold the ’79T’. But when there’s “only” a low straight penalty, you hold the ’79T’ when it’s a 3 or 4, but hold the ace when it’s a 2 or 5. What gives? I never noticed this before. That is what this article is about. If it’s already more technical than you want, I don’t mind if you skip the rest of the article.
The value of holding an ace by itself in this game comes from the following possible post-draw results:
W Royal Flush — 4000 coins
X ‘A2345′ straight flush — 250 coins
Y Flush — 30 coins
Z AKQJT straight — 20 coins
AA A2345 straight — 20 coins
BB AAAA where the fifth card is a 2-4 — 2000 coins
CC AAAA where the fifth card is a 5-K — 1000 coins
DD 2222-4444 where the fifth card is an ace — 1000 coins
EE A Quick Quad such as 3332A — 400 coins
FF A Quick Quad such as 6665A — 260 coins
GG Full house — 45 coins
HH Three of a kind — 15 coins
II Two pair — 5 coins
JJ Pair of aces, kings, queens, or jacks — 5 coins
So when you’re comparing A2’79T’ and A3’79T’, which of the above categories give different values to holding the ace by itself? (It has something to do with three 2s and four 3s left in the pack of 47 remaining cards compared to having four 2s and three 3s.) Answer this question and you’ll know why you hold the ace from the first and the ’79T’ from the second.
It’s a different problem to compare A5’79T’ with A4’79T’. But the answer lies in one of the above categories. (It has something to do with three 5s and four 4s left in the pack of forty-seven remaining cards compared to having four 5s and three 4s.) Answer this question and you’ll know why you hold the ace from the first and the ’79T’ from the second.
I strongly encourage you to work this out for yourself. You’ll learn more if you do. You’ve already been given some pretty big hints. If you can’t figure it out you’re in good company. I breezed right over this until I was reviewing for play on Easter Sunday. And I had previously looked at the WoO strategy calculator for this game dozens of times.
Let’s compare the case where there’s an off-suit 2 missing compared to an off-suit 3. Obviously this doesn’t affect any flush hand because the cards we’re talking about are off-suit. It doesn’t affect the A2345 straight, because a missing 2 has an effect equal to a missing 3. Going down the list, the only two categories I found that would be affected differently are EE (a 400-coin Quick Quad) and GG (a 45-coin full house.) If there are others, please let me know.
Let’s look at the total number of 222AA and 3332A Quick Quads when you are starting with a lone ace. With all other cards remaining in the pack, there are twelve ways to get 222AA because there are four ways to pick three 2s and you can get any of the three remaining aces. When a 2 is missing from the pack, that cuts this down to three available 222AA Quick Quads (out of the 178,385 possibilities). When a 3 is missing, all twelve of the 222AA Quick Quads are still possible.
There are sixty-four ways to draw a 3332A Quick Quad beginning with a single ace. You can get three 3s in four different ways, but there are also four 2s in the pack and three aces that can be mixed and matched in any way. Removing a 2 from the pack cuts this down to forty-eight different ways. Removing a 3 from the pack cuts this down to sixteen.
We also need to consider the hand 4443A when drawing four cards to a bare ace. When we have discarded a deuce, there are 16 ways to get this Quick Quad (four ways to get 444 and four ways to get a 3). When we have discarded a trey, there are only 12 ways to get this Quick Quad because there are now only three 3s remaining in the pack.
So when you’re looking at the total number of 222AA, 3332A, and 4443A possibilities, missing a 2 from the pack costs you nine 222AA Quick Quads, sixteen 3332A Quick Quads, and no 4443A for a total of twenty-five 400-coin hands. Missing a 3 from the pack costs you no 222AA Quick Quads but it costs you forty-eight 400-coin 3332A Quick Quads and four 4443A Quick Quads. Comparing those hands, it’s easy to see that missing a 3 hurts the value of an ace much more than missing a 2.
Now let’s look at drawing 222AA and 333AA full houses when beginning with a lone ace. 222AA is a Quick Quad, not a full house, so we’ve already taken that into consideration in the previous paragraph. When all cards are still available (other than the ace you are holding) there are four ways to get 333 and three different ways to draw the second ace — for a total of twelve full houses. If a 3 is discarded pre-draw, this is reduced to three different ways to get the 333AA full house — for a difference of nine 45-coin full houses.
So what is the cost difference between holding the lone ace and holding ’79T’? For a five-coin dollar player who is holding a lone ace, having a 3 missing from the pack costs 0.80 cents more than having a 2 missing from the pack. Clearly it’s a close play whether to hold ’79T’ or the single ace and adding 0.80 cents to the value of the ace is enough to make a difference in those plays.
The following is a much shorter, less technical explanation of why you hold the ace from A5’79T’ but not from A4’79T’. The difference comes down to the total number of 4443A and 5554A Quick Quads. Missing a 5 doesn’t hurt the chances for the 4443A Quick Quad at all. Missing a 4 hurts the chances of 4443A exactly as much as missing a 5 hurts the chances of 5554A, except the former costs you 400 coins and the latter costs you “only” 260 coins. Also, missing a 4 hurts the chances for the 5554A Quick Quad. In sum, missing a 4 hurts you quite a bit more than missing a 5.
As of the time I’m writing this, I haven’t devised a shorthand terminology for how to add these hands to my chart. I will, but it’s not a high priority. The hands we are talking about (e.g. A2’78T’ — A5’78T’) each appear only about once every 108,290 dealt hands and the differences are small. For me, though, it’s not just the size of the difference. I get a kick out of finding a new play I previously didn’t know. The research and thinking I put into any new play makes it easier to understand the next one.
Even though I’m not a fan of the WoO notation, this is a case where his methodology presented something that my methodology doesn’t even have the tools to describe. Yet. And for that I tip my hat.