Frank Kneeland and I are creating progressive strategies for a number of games which are now in the M casino. If you don’t know which games I’m talking about, see www.lasvegasadvisor.com/bob_dancer/2011/0510.cfm We are giving you a list of numbers saying that you should change your strategy if the royal flush is higher than a certain number. Today’s article discusses how we came up with those numbers.
Assume you are playing dollar 9/6 Jacks or Better, five coins at a time. You are dealt A♠ T♠ J♥ 6♣ 4♦. The proper play, of course, is to hold A♠ J♥. But now let’s assume you are playing the same game with a royal of $8,000 rather than $4,000. This will increase the value of A♠ T♠ but leave the value of A♠ J♥ as it was. Is A♠ J♥ still the right play, or is maybe A♠ T♠ correct now? And if it changed, when did it happen? How can you calculate this?
The amount of the royal flush where A♠ J♥ and A♠ T♠ are exactly equal in terms of expected value is called the Breakeven Point, or BP. To my knowledge, Stanford Wong in his Professional Video Poker was the first to use this term in print, although he abbreviated the term as BEP rather than the BP I prefer. Wong’s book was on 8-5 Jacks or Better Progressive, but the theory behind BP is the same in other video poker games as well.
BP analysis is typically done in terms of COINS, but in my opinion it’s a lot more useful when done in terms of DOLLARS AND CENTS. In this I am going to use dollar denominated games. 4,000 coins is, of course, $4,000. If you play for nickels, quarters, half-dollars, $2, $5 or higher, it is a simple matter to multiply the dollar BPs by the appropriate factor.
Notice the value of A♠ J♥ remains constant at $2.37. Since the A♠ and J♥ are of different suits, they can never be part of the same royal flush. No flush or straight flush figures into the value of A♠ J♥, but the probability of high pairs, two pairs, 3-of-a-kinds, straights, full houses, and 4-of-a-kinds is considered.
The slanted line represents the value of A♠ T♠. With a royal flush of $4,000, A♠ T♠ has a $EV of $2.25. That is lower than the $2.37, so we should hold A♠ J♥ when we are playing for a royal of $4,000. At a royal of $8,000, the $EV of A♠ T♠ is $2.49. This is quite a bit above $2.37, so at this royal flush value we should play A♠ T♠.
The place where the two lines cross is the BP. That is, the value of the royal where the $EV of A♠ J♥ was equal to the $EV of A♠ T♠. The change in the royal value of this turns out to be $1,975 (for a total royal flush amount of $5,875), a figure we will soon derive.
The graph is useful to show that the value of A♠ T♠ increases while the value of A♠ J♥ holds constant and sooner or later they must be equal. But we don’t need the graph to know what the $EV will be when the two lines meet. We know the $EV will be $2.37 at that point, because the $EV of A♠ J♥ is ALWAYS $2.37. What we need is a formula to determine what the value of the royal flush will be when the $EV of A♠ T♠ is $2.37.
The formula is not very difficult. It is, simply, ΔRF = C x Δ$EV. Of course the symbols all need to be defined, so let’s do that now. Does the funny looking triangle look like Greek to you? It should. IT IS GREEK! Actually it is the capital Greek letter “delta”, which is the symbol mathematicians usually use to indicate “the change in”. So ï�„RF simply means the change in the value of the royal flush, and ï�„$EV means the change in the value of the dollar expected value.
The symbol C stands for the number of possible combinations. “Combination” is an everyday term, but it also has a precise mathematical definition. Here we mean the number of unique draws we could make from a starting position, IF ORDER DOESN’T MATTER. That is, if we draw three cards (like we are doing in this example), the draw A♣ 5♥ 3♠ is considered exactly equivalent to A♣ 3♠ 5♥, or 3♠ A♣ 5♥ or any of the three additional permutations. These are different combinations than A♥ 3♥ 5♠ or A♥ 3♦ 5♠. The possible values of C will be:
| 178,365 for a 4-card draw |
| 16,215 for a 3-card draw |
| 1,081 for a 2-card draw |
| 47 for a 1-card draw |
Regular readers of my writing are familiar with these numbers. For a 52-card game, these numbers remain constant.
The easiest way to get $EV is to use a video poker computer trainer. Using Video Poker for Winners, go to 9/6 Jacks or Better with a royal flush of 4000. Make sure you are on “max bet”, and then go to ANALYZE → SELECT SPECIFIC CARDS.
Enter the hand A♠ T♠ J♥ 6♣ 4♦ and look at the numbers. A♠ J♥ appears on the top line, and the first number you see is 2.37155. This is the $EV. A♠ T♠ appears a few lines lower, and the first number you see there is 2.24975. In the graph I rounded these numbers off to 2.37 and 2.25, but for BP analysis we want to use the extra significant digits.
Since we are drawing three cards to A♠ T♠, our formula now becomes
| ΔRF = 16,215 x (2.37155 — 2.24975) |
| ΔRF = 16,215 x (0.12180) |
| ΔRF = 1975 (rounded) |
This means we add $1,975 to the original $4,000 and come up with a BP of $5,975. When the royal flush is lower than $5,975, hold A♠ J♥. When it is above $5,975, hold A♠ T♠. When it’s exactly $5,975, take your pick. (For practical purposes, since it pleases most people to hit a royal flush, when the numbers are tied, I recommend trying for the royal. It has a 1-in-16,215 chance of making you very happy.)
If you don’t trust your math, go back to Video Poker for Winners and change the pay schedule of 9/6 Jacks so that 5975 is the amount of the 5-coin royal. If you do that, and enter the A♠ T♠ J♥ 6♣ 4♦ hand we’ve been talking about, you’ll see that both A♠ J♥ and A♠ T♠ have the same value, and it is $2.37155.
Let me give you another example. I urge you to work it through before I give the explanation. The only way for you to understand it is to do it.
The example hand is K♥ Q♥ J♥ J♠ 5♦. At a royal flush of $4,000 it is correct to play J♥ J♠. We want to know the value of the royal flush where you should hold K♥ Q♥ J♥. The relevant $EVs are 7.6827 for J♥ J♠ and 7.4422 for K♥ Q♥ J♥.
The reason I am changing the subject now is that I urge you to go try this one and I don’t want you to accidentally see the answer.
Calculating BPs is tedious. And there are a lot more of them than you might think. Our first example seemed easy enough, but the number would have been different if the 6♣ had been the 6♠. The $EV of A♠ J♥ wouldn’t have changed, but the $EV of A♠ T♠ would be LOWER because there are fewer flush opportunities once the 6♠ is thrown away. (An important concept in this process which isn’t obvious to some beginners is when you throw away a hand before the draw, it’s impossible for that same card to come back after the draw.)
Okay. Let’s get back to our problem. Since this will be a 2-card draw (for the K♥ Q♥ J♥), the appropriate value for C is 1,081.
F
| ΔR | = | C x Δ$EV |
| = | 1,081 x (7.6827 — 7.4422) | |
| = | 1,081 x (0.2405) | |
| = | 260 (rounded) | |
| So the BP | = | $4,000 + $260 = $4,260 |
Did you get it correct? It’s not tough, but it does take some practice to get good at it. (It’s impressive to me that back when Frank was managing the video poker team, they did all of this without computers. When computer trainers became common, it became relatively easy for players to figure this stuff out for themselves, which gave them less incentive to belong to a team where the information was presented to them.) There are some additional considerations in BP analysis that I’m saving for next time, but you now have the basics that will take you through 99+% of the hands.