How Do You Figure?

There are a variety of promotions that appear from time to time. To analyze them, you need to be able to figure out how much they are worth. You can wait until someone posts an estimate of what a promotionBob Dancer is one of the world's foremost video poker experts.  He is a regular columnist for Casino Player, Strictly Slots, and the Las Vegas Review-Journa land has written an autobiography and a novel about gambling.   He provides advice for tens of thousands of casino enthusiasts looking to play video poker.  Bob's website is www.bobdancer.com  is worth on the Internet (if that ever happens), or you can learn to figure it out yourself.

I've always preferred the "figure it out yourself" approach, for a number of reasons. First, of course, I know how to figure out many of these promotions. If I didn't know how, this wouldn't be an option. Second, many of the best promotions are those that don't become well-known. By the time the information is posted, other players (and casino managers) read the posts and either it's very difficult to get a seat or the managers realize that they are offering too much and kill the deal.

The promotions I'm going to talk about today deal with the theory of combinations. Combinations are how many different ways something can happen when order isn't important. Although mathematicians use different symbols, I'm going to use 52-C-5 to indicate a combination of five things taken out of a pool of 52. That would be the number of different 5-card hands that could be dealt from a 52-card deck. The formula is

Number of Combinations for 52-C-5 = 52! / 5! (52 ­ 5)!

The exclamation point is called "factorial". 52! would be 52 x 51 x 50 x . . x 3 x 2 x 1. Noting that (52 ­ 5)! is 47!, the last 47 terms of the numerator are cancelled out by 47!. Also 5! = 5 x 4 x 3 x 2 x 1 = 120. This leaves us with 52-C-5 = 52 x 51 x 50 x 49 x 48 / 120 = 2,598,960.

When we are talking about DEALT hands, each of these almost 2.6 million hands are equally likely. When we are talking about RESULTING hands after the play, the hands are no longer equally likely because now strategy is important.

If we are talking about a problem such as "how often are 4-aces dealt", we use the formula 4-C-4 x 48-C-1 / 52-C-5. 4-C-4 is the number of different ways you can draw 4 aces out of a total of 4 (which is obviously 1), and 48-C-1 are the number of ways a fifth card can be drawn out of the 48 cards remaining after the aces are selected. That number is, obviously, 48. So the probability that we draw four aces is 48 / 2,598,960 = 1 / 54,145. That means that once every 54,145 hands, you'll be dealt four aces, or four sevens, or any other specific quad.

Let's assume a casino was offering a 500-coin bonus if you were dealt aces, assuming you were playing five coins. How much would that be worth in percentages? This would be 500 / 54,145 x 5 = .0018 = 0.18%. Whether this was worth playing, of course, would depend on what the best game at the casino, plus the slot club, plus the other promotions. If, for example, 8/5 Jacks or Better (97.3%) is the best game, this promotion wouldn't make the game desirable. If the game were NSU Deuces Wild (99.73%) with a .25% slot club, that would be worth 99.73% + 0.18% + 0.25% = 100.16%, which may or may not be worth playing depending on what ELSE was available. (For quarter players, you can do better with full pay deuces wild without any slot club. For $10 players, pull up a chair. This is a decent game for those stakes.)

So how much is 0.18% in "real money"? Assuming you are playing 600 hands per hour, we're talking about $1.385 per hour for a quarter player, $5.54 per hour for the dollar player, etc. Most hours, of course, it will end up adding zero, buy once every 90 hours (i.e., 54,145 / 500), you'll collect $125 if you're a quarter player or $500 if you're a dollar player.

Let's try another. How often is the 77744 full house dealt? We have 4-C-3 x 4-C-2 / 52-C-5 = 4 x 6 / 2,598,960 = 1/108,290. 4-C-3 = 4 is the number of ways three 7s can be chosen from the four possible (i.e. 7c7d7h, 7c7d7s, 7c7h7s, and 7d7h7s) and 4-C-2 = 6 is the number of ways two 4s can be selected from the four possible (i.e. 4c4d, 4c4h, 4c4s, 4d4h, 4d4s, 4h4s).

I was surprised the first time I learned that a specific dealt quad was twice as likely to occur than a dealt specific full house. So a 1,000-coin bonus on a dealt JJJ55 (or any other specific full house) would be worth the same 0.18% that a 500-coin bonus on quad fours (or any other specific 4-of-a-kind). Receiving twice the bonus half as often comes out to the same percentage.

At the current time, I don't know of any promotions on dealt hands, but we see them from time to time. Coast casinos once offered a double pay for max-coin dealt royals. Using the procedures discussed in this article, you should be able to determine that that was worth 0.123%.