"I had to see the flop"

The first full hand I watched play out after taking my seat at the Rio tonight had Player A limp in, Player B raise to $15, Player A call. After the flop, it was check/bet/fold. Player A showed his cards to the table before folding. "I had the jack-ten of hearts. I had to call. I had to at least see the flop, for a possible $4000."

I hadn't been aware of it, but apparently the Rio high-hand jackpot is up to a little over $4000 for hitting a royal flush.

Let's run a few numbers. If you start with any two of the five cards needed for a royal flush, how likely is it that you will have the royal if you see the hand out to the bitter end? Obviously, you're going to need the final board to have the other three exact cards that you need, leaving two other slots that can be filled with any ol' cards. So after we specify what five of the seven cards have to be, there are 47 possibilities for one of the open slots, and then 46 for the final slot, for a total of 2162 different final boards that will complete our hand (assuming that we don't care about what order they come in).

The total number of boards that we might get, after eliminating our two parts of the royal from the deck (because they are in our hand), is denoted by (50, 5). Running this on Excel, that turns out to be exactly 2,118,760. In other words, of a little over two million possible five-card boards that might show up, a little over 2000 of them will give us the royal flush. The odds are 980:1. Let's round it to 1000:1 to make things neat and simple.

With a $4000 jackpot, the expected value is therefore about $4 every time you have two suited Broadway cards and see the flop. Mathematically, then, you should imagine that the house has plopped an extra $4 into the pot as soon as you see those two hole cards, and then make your decisions with that slightly altered assessment of what the pot is offering you.

From the discussion that ensued after this guy folded, it was clear that he dislikes playing J-10, and called the $15 raise only because of the jackpot possibility. In other words, he threw in an extra $12 (beyond the $3 call he had already made) in order to win $4.

Actually, the situation is worse than that, because that extra $12 only gets us to the flop, and even when the royal is destined to come, it will most often not all hit at once like that. So we're going to have to be willing to invest even more money if we get one or two more pieces of the royal flush on the flop and an opponent bets. The real amount that one would have to invest in order to collect that hypothetical $4, playing the way this guy was apparently willing to pay, would certainly be, on average, something like two or three or four (or more) times what we've already calculated. On the other hand, sometimes he will win the pot even when he doesn't win the jackpot, so that reduces his average price (or you can say that it increases his average profit) some. For a hand that he recognizes isn't intrinsically strong, this surely won't be enough to offset his entire entry price, but let's be extremely generous and assume that he'll win often enough to compensate for what he'll have to pay, on average, on the later streets, thus leaving him with just the pay-12-to-win-4 deal.

Of course, he would protest that he didn't spend $12 in order to win $4; he spent $12 in order to win $4000. OK, let's look at it from that point of view. How much will he have to spend, on average, in order to collect that $4000, if this situation were to repeat itself endlessly? That's easy: about $12,000 ($12 each time, for 1000 iterations). And, again, this is the best-case scenario--if the royal comes on the flop or our opponent is generous enough to let us peel off two more free cards, which doesn't seem like the most common way it will play out.

By his own words and actions, we can infer that this guy is willing to spend $12 (or more) to win $4, or $12,000 (or more) to win $4000. He can pick either of those ways of expressing it that he likes, and I won't argue with him. But, of course, they're both equally stupid.


One of the things I would most like to see in one of the poker publications is a series of columns or articles with rigorous mathematical assessments of various poker room promotions, how they distort the pot odds, and to what degree should one alter one's play to adjust for them?

For example, a casino I used to play at in Minnesota had an "aces cracked" promotion one day a week--you'd get a rack of white chips ($100) if your pocket aces got beaten. Some players insisted that the only smart way to play aces was to limp, hope for lots of opponents, play completely passively and see the hand to the end as cheaply as possible, hoping to get beaten and pick up the $100. The problem with this strategy is that often the aces would still hold up, and you're left without the bonus, and with a tiny pot. Other players insisted that because of the problems with that strategy, it was best to play the hand as usual, hope and expect to usually win the pot, and let the $100 bonus just be sort of a net to break your fall if you got unlucky. Arguments about this took place every single week. I'd love to see somebody take reasonable ranges of assumptions about the variables involved (number of callers, size of the pot, probability of holding up against X opponents, etc.) and crank out a mathematically sound answer. I sure can't do any more than guess at what it is.

I think that bad-beat jackpots probably introduced relatively little distortion, because the threshold requirements are usually two big hands that would inevitably get played out anyway. It's hard to imagine folding four of a kind just because an opponent with the perfect two cards in his hand might have a straight flush, for example.

But there undoubtedly is a mathematically optimal way to account for various promotions--such as should I hold on to a J-5 of diamonds that I would otherwise fold, for the possibility of hitting a diamond flush at the Palms? In order to answer that, you'd have to know approximately how many people hit one every day, so that you could calculate the expected bonus value of hitting the flush, then make some estimate of how much EV you're losing in the pot by calling. Those aren't easy or obvious numbers to come by.

I have written to a few of the more mathematically minded columnists at Card Player and suggested this topic, but none of them have taken up the challenge yet. Oh well. I have a pretty good idea that the guy at saw at the Rio tonight would flip right past them, or would get a glassy look in his eyes at all the numbers, then say, "But I might win $4000!" (Think of the look and tone of the guy in "Spinal Tap" saying "But these go to 11!")