{"id":995,"date":"2011-05-24T20:04:14","date_gmt":"2011-05-24T20:04:14","guid":{"rendered":"http:\/\/gwae.apps-1and1.com\/?p=995"},"modified":"2011-05-24T20:04:14","modified_gmt":"2011-05-24T20:04:14","slug":"how-to-calculate-breakeven-points","status":"publish","type":"post","link":"https:\/\/www.lasvegasadvisor.com\/gambling-with-an-edge\/how-to-calculate-breakeven-points\/","title":{"rendered":"How to Calculate Breakeven Points"},"content":{"rendered":"

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<\/a> 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.<\/p>\n

Assume you are playing dollar 9\/6 Jacks or Better, five coins at a time. You are dealt A\u2660 T\u2660 J\u2665 6\u2663 4\u2666. The proper play, of course, is to hold A\u2660 J\u2665. 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\u2660 T\u2660 but leave the value of A\u2660 J\u2665 as it was. Is A\u2660 J\u2665 still the right play, or is maybe A\u2660 T\u2660 correct now? And if it changed, when did it happen? How can you calculate this?<\/p>\n

The amount of the royal flush where A\u2660 J\u2665 and A\u2660 T\u2660 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.<\/p>\n

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.<\/p>\n

\"\"<\/center>Let’s start off with the graph. The graph shows how the value of A\u2660 J\u2665 and A\u2660 T\u2660 vary as the value of the royal flush increases. The vertical axis shows “dollar expected value” (abbreviated $EV), which is the value, in dollars and cents, of each combination when you are playing the game with five $1 coins. The horizontal axis shows the change in value of the royal flush as we increase from $4,000, denominated in hundred dollar units. The far left edge of the graph represents a royal of 4,000 and the far right edge of the graph represents a royal of 8,000.<\/p>\n

Notice the value of A\u2660 J\u2665 remains constant at $2.37. Since the A\u2660 and J\u2665 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\u2660 J\u2665, but the probability of high pairs, two pairs, 3-of-a-kinds, straights, full houses, and 4-of-a-kinds is considered.<\/p>\n

The slanted line represents the value of A\u2660 T\u2660. With a royal flush of $4,000, A\u2660 T\u2660 has a $EV of $2.25. That is lower than the $2.37, so we should hold A\u2660 J\u2665 when we are playing for a royal of $4,000. At a royal of $8,000, the $EV of A\u2660 T\u2660 is $2.49. This is quite a bit above $2.37, so at this royal flush value we should play A\u2660 T\u2660.<\/p>\n

The place where the two lines cross is the BP. That is, the value of the royal where the $EV of A\u2660 J\u2665 was equal to the $EV of A\u2660 T\u2660. 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.<\/p>\n

The graph is useful to show that the value of A\u2660 T\u2660 increases while the value of A\u2660 J\u2665 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\u2660 J\u2665 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\u2660 T\u2660 is $2.37.<\/p>\n

The formula is not very difficult. It is, simply, \u0394RF = C x \u0394$EV. Of course the symbols all need to be defined, so let’s do that now. Does the funny looking triangle look like Greek to you? It should. IT IS GREEK! Actually it is the capital Greek letter “delta”, which is the symbol mathematicians usually use to indicate “the change in”. So \u00ef\ufffd\u201eRF simply means the change in the value of the royal flush, and \u00ef\ufffd\u201e$EV means the change in the value of the dollar expected value.<\/p>\n

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\u2663 5\u2665 3\u2660 is considered exactly equivalent to A\u2663 3\u2660 5\u2665, or 3\u2660 A\u2663 5\u2665 or any of the three additional permutations. These are different combinations than A\u2665 3\u2665 5\u2660 or A\u2665 3\u2666 5\u2660. The possible values of C will be:<\/p>\n\n\n\n\n\n\n
178,365 for a 4-card draw<\/td>\n<\/tr>\n
16,215 for a 3-card draw<\/td>\n<\/tr>\n
1,081 for a 2-card draw<\/td>\n<\/tr>\n
47 for a 1-card draw<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Regular readers of my writing are familiar with these numbers. For a 52-card game, these numbers remain constant.<\/p>\n

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 \u2192 SELECT SPECIFIC CARDS.<\/p>\n

Enter the hand A\u2660 T\u2660 J\u2665 6\u2663 4\u2666 and look at the numbers. A\u2660 J\u2665 appears on the top line, and the first number you see is 2.37155. This is the $EV. A\u2660 T\u2660 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.<\/p>\n

Since we are drawing three cards to A\u2660 T\u2660, our formula now becomes<\/p>\n\n\n\n\n\n
\u0394RF = 16,215 x (2.37155 \u2014 2.24975)<\/td>\n<\/tr>\n
\u0394RF = 16,215 x (0.12180)<\/td>\n<\/tr>\n
\u0394RF = 1975 (rounded)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

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\u2660 J\u2665. When it is above $5,975, hold A\u2660 T\u2660. 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.)<\/p>\n

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\u2660 T\u2660 J\u2665 6\u2663 4\u2666 hand we’ve been talking about, you’ll see that both A\u2660 J\u2665 and A\u2660 T\u2660 have the same value, and it is $2.37155.<\/p>\n

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.<\/p>\n

The example hand is K\u2665 Q\u2665 J\u2665 J\u2660 5\u2666. At a royal flush of $4,000 it is correct to play J\u2665 J\u2660. We want to know the value of the royal flush where you should hold K\u2665 Q\u2665 J\u2665. The relevant $EVs are 7.6827 for J\u2665 J\u2660 and 7.4422 for K\u2665 Q\u2665 J\u2665.<\/p>\n

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.<\/p>\n

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\u2663 had been the 6\u2660. The $EV of A\u2660 J\u2665 wouldn’t have changed, but the $EV of A\u2660 T\u2660 would be LOWER because there are fewer flush opportunities once the 6\u2660 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.)<\/p>\n

Okay. Let’s get back to our problem. Since this will be a 2-card draw (for the K\u2665 Q\u2665 J\u2665), the appropriate value for C is 1,081.<\/p>\n

F<\/p>\n\n\n\n\n\n\n\n
\u0394R<\/td>\n=<\/td>\nC x \u0394$EV<\/td>\n<\/tr>\n
<\/td>\n=<\/td>\n1,081 x (7.6827 \u2014 7.4422)<\/td>\n<\/tr>\n
<\/td>\n=<\/td>\n1,081 x (0.2405)<\/td>\n<\/tr>\n
<\/td>\n=<\/td>\n260 (rounded)<\/td>\n<\/tr>\n
\u00a0So the BP<\/td>\n=<\/td>\n$4,000 + $260 = $4,260<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

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.<\/p>\n\"Facebook\"<\/a>\"twitter\"<\/a>\"youtube\"<\/a>\"instagram\"<\/a>","protected":false},"excerpt":{"rendered":"

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 […]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_exactmetrics_skip_tracking":false,"_exactmetrics_sitenote_active":false,"_exactmetrics_sitenote_note":"","_exactmetrics_sitenote_category":0,"spay_email":"","jetpack_publicize_message":"","jetpack_is_tweetstorm":false},"categories":[6],"tags":[],"yoast_head":"\nHow to Calculate Breakeven Points - Gambling With An Edge<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.lasvegasadvisor.com\/gambling-with-an-edge\/how-to-calculate-breakeven-points\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"How to Calculate Breakeven Points - Gambling With An Edge\" \/>\n<meta property=\"og:description\" content=\"Frank Kneeland and I are creating progressive strategies for a number of games which are now in the M casino. 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