The GameMaster's Poker School
Lesson 4 - Poker Math: Longshots

The other day I was dealt an exciting hand that got me thinking about longshots - both the kind you want to hit and the kind to avoid - so this lesson is the result. I was dealt the Ace and Queen of spades as my pocket cards in a game of Hold 'em and, as you hopefully know by now, it's a pretty good way to start. But it got even better when the flop came: Ks, Js, 9d. Now I had a 4-card Royal Flush and needed to catch only the 10 of spades to complete it. "Only" is a relative word, of course because the odds against me catching the 10 of spades on the turn was 1 in 47 (I've seen 5 cards to this point, so 47 are left and only one of them is the 10 of spades). Of course, any 10 would make a Straight and any spade would make a Flush, but darn it, I wanted the Royal!

My interest in completing the Royal was not just ego-driven, because the casino where I was playing offers a bonus to anyone who finishes a hand with a Royal. So, not only was I guaranteed to win the pot for the hand (you don't even have to worry about a tie with a Royal), but I'd also get $500 thrown in as well. I knew the odds against me making it were huge, I've drawn to enough 4-card Royals at Video Poker to know that, but at least here I had two shots at it - one on the turn and one on the river. Because it wouldn't matter when I got the card, only if I got it, I started to think about, first, what I was going to buy with the $500 (I 'm an optimistic rascal) and second, what kind of expected value is added to our poker hands by such bonuses?

You all remember "expected value" (EV), right? It's a mathematical calculation based upon what will happen over many hands of play in the case of poker and Blackjack. In other words, we won't always win with pocket Aces in Hold 'em, but over thousands of hands we'll win enough so that we can put a value on it. For example, if we win with AA 50% of the time, on average, then this starting hand has an EV of 50% of all the $$$ we bet in that situation. Of course we can't pin down the exact size of our bets because it'll be different from hand to hand, although over a period of time we can probably come up with a fairly accurate average number. But in the case of a Royal Flush bonus, we know it's a fixed amount so all we have to do is calculate how often we'll get one and that'll give us an EV per hand.

Why is it important to know how much EV is added to each hand by a Royal Flush bonus? Well, it isn't really, but it's a simple calculation, so why not? Every little bit helps, you know, especially when you're starting out. Combine bonuses like these with the fact that most online poker rooms have fairly low rakes (compared to brick-and-mortar card rooms), plus there's no dealer to tip and you have a definite leg-up over your "real-life" counterpart. If nothing else, the cost of gaining some experience at poker will be somewhat lower if you do it online rather than at a brick-and-mortar card room. But I digress.

Just what's a Royal bonus worth, anyway? To figure it on a per-hand basis, we need to calculate the probability of getting a Royal and that will tell us how often we can expect, on average, to get one. To draw a Video Poker analogy here, we know that a Royal will occur, on average, about once every 40,000 hands in a 9/6 Jacks or Better game, which means the probability is 1 divided by 40,000 = 0.000025. Because that Royal will usually pay 800 for 1, it means that Royal Flushes add .000025 x 800 = 0.02 or 2% to the total return of a 9/6 Jacks game, which is 99.54%. In other words, if there were no Royal "bonus" in a Jacks VP game, the return would be only 97.54%. So does that mean we should expect to get a Royal once every 40,000 hands at Hold 'em poker? Sadly no, because of the way the game is structured. At Video Poker, you are dealt 5 cards, may hold or fold any or all and then are dealt replacement cards, so you have a "universe" of 10 cards from which to make your Royal.

In Hold 'em, you are dealt 2 pocket cards that you must keep if you want to keep playing the hand, then 5 more cards come if the hand is played to the end. The universe here is obviously only 7 cards, so it's probably not too difficult to imagine that we can't expect to get a Royal once every 40,000 hands. However, there is more than one way to make a Royal in Hold 'em, just as there is in Video Poker. The first of those is to get a Royal dealt to you. This can happen at VP because you receive a 5-card hand and the probability of that occurring is 1 in 649,740. Well, the same thing can happen at Hold 'em, because you can be dealt two suited Royal Flush cards in the pocket and then the flop can fill your Royal. The odds of that happening are exactly the same as getting one on the deal in Video Poker:1 in 649,740. Talk about long shots, eh?

But don't dispair because there's a much more common way for it to happen and that's to have the Royal unfold like the one I had. Two suited Royal cards in the pocket, two on the flop and then draw the fifth on either the turn or the river. I'll spare you the background math, but the probability of being dealt two suited Royal Flush cards is 1 in 33 (33.15 to be exact), then getting two of the three you need on the flop is 1 in 139 and finally, getting the 5th card on either the turn or river is 1 in 23 (23.25 to be exact). Multiply those three together: 33.15 x 139 x 23.25 and you get 1 in 107,133, which you can safely round to 1 in 100,000. If you'll receive a $500 bonus for hitting a Royal, you can expect it to happen about once every 100,000 hands, so it's worth $500 divided by 100,000 = $.005 or about a half-cent per hand.

So, how did my hand work out? If you remember, I had A-Q spades in the pocket, the flop came Ks, Js, 9d, so all I needed was the 10s. The turn was 3d, the river was 3h and I lost to a player holding Kc, 3s. Yep, he had a Full House and I had a busted Flush. Hey, that's how it is in poker sometimes. Don't worry, I'll get over it, so lets talk about some other longshots.

These are the type of longshots to avoid. Or, if you won't avoid the situation, at least make sure that the "pot odds" are rewarding you. In Lesson 2, I presented a chart of the various odds of completing a hand, such as a 4-card Flush and so forth. The hands presented there were the types of hands you'll run into all the time, unlike the Royal Flush we discussed earlier. The hands I'm going to discuss here are also the type you'll run into a lot, but in most cases you shouldn't play them and the numbers will show you why. For example, you may find yourself with some pretty nice pocket cards like Ah,10h and the flop comes 2s, 6d, 7h. You don't have much, other than a 3-card Flush draw and a double-inside Straight draw. But, were you to get the Flush, it would be the "nuts" and would beat any Straight that forms. But, with 2 cards to come, can you get what's called a "runner-runner" to fill the hand? Certainly that's possible, but the exact odds of success are pretty much against it happening, so you can waste a lot of $$$ in trying. Meantime, the guy with pocket Kings is betting every round and unless another Ace falls, he's probably going to win the pot.

If you have a 3-card Flush, that means there are 10 cards of that suit remaining in the deck (remember that we don't count anything we can't see, so even though other players may also have cards of that suit, they don't matter for purposes of calculating our odds). So, with 10 cards of the remaining 47 (52 minus the 2 pocket cards, minus the 3 cards on the flop) being cards that will help us and two chances to get them, it doesn't seem like too bad a deal. But don't forget that both of the last two cards have to be hearts (in this example) or we'll have a hand worth basically nothing. Sure, you might win with an Ace-high, but don't bet on it. Literally.

Nope, we need to hit two running hearts for this to work and the odds against that happening are an amazing 24 to 1. Believe me, I had to double-check my figures when I got that number because it seemed just too high to be correct, but it is. The quick mathematical solution is to figure the probability of getting a heart on the turn (10/47) or 0.212 and multiplying that by the probability of getting a heart on the river (9/46) or 0.195. Well, multiply 0.212 by 0.195 and you get 0.0415. Remember how I showed you to convert probability to odds in Lesson 2? First, subtract the probability of 4 from 100 and you'll get 96. Now divide 96 by 4 and you'll get 24 to 1 as the odds against. This obviously means that the value of the pot at the flop is going to have to be 25 times the bet you have to make in order for it to have a positive expectation. I've seen such a thing, but it's very rare, so most of the time you should be folding your 3-card Flushes.

Now I realize there may be other reasons for staying with the hand, but the odds against making various hands that I outlined in Lesson 2 will guide you there. And certainly, if you had the same pocket cards but the Ace were a Jack, then "fuhgedaboudit", because you wouldn't be drawing to the "nut" Flush. Yet, a lot of players, particularly in low-limit games, will cling to a "suited" Ace (an Ace plus any card of the same suit) in the pocket until the bitter end. Don't forget this: A dollar you don't lose is a dollar earned. The object of this lesson is to cut down on the number of long-shot bets that we all make from time-to-time. Don't get me wrong; if the pot odds are there, go for it. But if they're not, then fold.

Okay, enough preaching. Here is a list of various hands you might find yourself with after the flop. In other words, you've seen five cards, two are yet to come and now you have to make a decision to bet or fold. This chart is really just a continuation of the chart I presented in Lesson 2: 

Notes and comments: I've included the Ace-high hands because I've seen so many players hold onto their Aces with a death-grip, as I mentioned above. Now don't get me wrong; Trip Aces will win most hands of Hold 'em, but as this chart shows you it'll happen only once every 33 times you hold a single Ace at the flop. For me, this type of chart removes the guesswork, "intuition" or whatever you care to call it, from the game. If the pot odds warrant the play, do it, otherwise fold. Oh, I fully realize that the first time you fold a 3-card Flush, the turn and river will bring the cards you needed, but that'll be the exception, I assure you. As a quick review, the "Bet Multiplier" is something I presented in Lesson 2 and it's a quick way to see if the bet you must make to stay in the hand has a positive EV. In a $1/$2 game, for example, if the bet you must make to stay in the hand is $1 and you're drawing to a 3-card Flush, the pot should be at least $25. If the bet you must make is $2, the pot has to be $50 or you should fold.

Okay, get your homework and this'll do it until next time.

Homework

• Integrate the chart above with the chart of more reasonable probabilities that I presented in Lesson 2 and keep it near you as you're playing. You should play only when you have the edge and it's the pot odds that determine whether or not you do, so try to memorize as much of this as possible.
• If you're playing online, continue using the minimum starting hands chart that I presented in Lesson 3 and keep the combined chart of probabilities nearby as well. What you're doing with these is forming a solid foundation for your play that we'll expand upon in future lessons.
• A good book to get for poker math is "Hold 'em's Odds Book" by Mike Petriv. It sells for $24.95 and is available at www.conjelco.com/