Understanding the math behind Video Poker
By John Grochowski
Many questions from video poker players concern theories about certain hands being “due.” Some examples include the following:
“I have played for 10 years and never had a royal flush. Surely I should have had several by now. I certainly must be due.”
“I read somewhere that a video poker player should get a straight flush once every 9,000 to 10,000 hands. I have played for three years and never gotten a single straight flush, much less a royal flush. Am I not due for one?”
Other letters have more general concerns about how random games work. The following emails are examples of this:
“Isn’t it true that I should complete a flush once every five times that I am dealt four cards of a flush? It sure doesn’t seem like I complete them anywhere near that often.”
“I’ve read that a player should fill a four-card open straight flush once out of every 23 and a half times. It seems I never complete a single one of these.”
It is indeed hard to put what the math states into perspective while playing video poker. There are many reasons for this. The main one is likely that memory has the tendency to be highly selective.
Sure we remember when we hit a royal flush. That’s because this is the ultimate hand. If playing at a denomination of 50 cents or higher it is even more memorable. All play stops. The machine locks up. A slot attendant must hand pay your winnings and reset the machine. Yes, this very infrequent hand is definitely remembered.
What may not be remembered as clearly is the number of hands since the last occurrence of a royal flush. Unless the player takes some time to keep track of the number of hands played, the estimate will likely be grossly overstated.
As human beings we tend to remember the times when we go hand after hand after hand without any decent-paying hands. We remember this because we have to repeatedly feed more cash into the game. This is very unpleasant and it definitely makes an impression on us.
On the other hand, when we are getting some decent hands such as flushes, straights and full houses scattered among the other lower-paying (and non-paying) hands and the credit balance merely fluctuates between two non-memorable limits (such as a low of 40 and a high of 150), the hours can fly by. There is never a time when we are in danger of needing to feed more into the game, nor is there an instance when we are tempted to cash out a significant win. In essence, we are playing on autopilot.
In this case we tend to forget those three-of-a-kinds, straights, flushes and full houses that occur, but they are there.
There is a way, however, to actually see some of the mathematical probabilities playing out—at least for some of the more frequently occurring hands.
While not nearly as popular as it has been in the past, 100-play video poker is an excellent way for a video poker player to observe mathematical probabilities playing out. While it may be an excellent method of seeing the probabilities, one of the problems with 100-play is the cost per hand. Even at the minimum denomination of 1 cent, each 100-hand play costs a minimum $5 (5 cents per hand for 100 hands). This is the same as a dollar game and for many recreational video poker players it is too much. That is fine, if you cannot comfortably afford 100-play, don’t play it. Another problem to some is the fact that a single royal only pays $40—although there could be several of them per 100-play.
But assuming that you can afford this level of play and the low-paying royal flush doesn’t deter you, the 100-play game will dramatically show how random games work. The math for a full-pay Jacks or Better game determines the following hand frequency and probability:
TABLE 1: Full-Pay Jacks Or Better
|Occurs Approx. Every
|Four of a Kind
|Three of a Kind
|Jacks or Better
The probability column shows the percent probability. Since we are playing a 100-play game, you should receive, on average, the whole number part of the probability in each 100-play hand. That means you should expect an average of 55 hands that pay nothing for every 100-play hand. You should also expect 21 high pairs, 13 two-pair hands, seven three-of-a-kind hands, one straight, one flush and one full house. These hands are an average of all hands (no matter what cards are saved) over many hands and as such may be a little hard to actually see.
What is more apparent is the appearance of completed hands when several cards of the hand are held. Let’s look at some examples.
TABLE 2: Complete Hands
|Approx. Times / 100
|Four of an Inside Straight Flush
|1 of 47 = 2
|Four of an Open Straight Flush
|2 of 47 = 4
|Four of an Inside Straight
|4 of 47 = 8+
|Four of an Open Straight
|8 of 47 = 17
|46 of 1,081 = 4
|66 of 1,081 = 6
|969 of 1,081 = 90
Here are some things to consider:
- When holding four cards of an inside straight there are only four cards in the remaining 47 cards (52 minus the 5 dealt) that will complete the hand.
- When holding four cards of an open straight there are four cards at the low end and four cards at the high end of the open straight (a total of eight) in the remaining 47 cards that will complete the hand.
- When holding four cards of an inside straight flush there is one card in the remaining 47 cards that will complete the hand.
- When holding four cards of an open straight flush there are two cards (one at the high end and one at the low end) of the 47 cards that will complete the hand.
- When holding three of a kind there are 1,081 possible permutations of the hand. They are distributed as shown in the table above.
- Or, stated differently for the four-of-a-kind: There is one card out of the 47 remaining that will make a four-of-a-kind, but there are two possible positions, so there is a one in 23.5 chance of completing the four-of-a-kind.
The higher frequency of occurrence hands will dramatically show in the results of a 100-play hand so the player can get a good sense of how often the different hands actually occur. It is perfectly clear that the winning hands do not occur in any smooth sequence, but when adding up the total number, they tend to be very close to what the math of the game says.
You may want to try it sometime just to satisfy your curiosity.