Expected Values of Video Poker Hands

Video poker hands are paid based on the level of difficulty of making the hand.  How the hands are paid off is disclosed on the face of the video poker machine.  By using this payoff information and by knowing just how difficult it is to make any particular hand, we can evaluate how to correctly play any hand.

Let's consider again the hand dealt us of: 5©6©7©8§8¨.  This hand is either a low pair, a four card straight, a three-card straight flush or we could even discard all of the cards and draw five new ones.

To evaluate which hand to pursue, we must first know which version of video poker we are playing.   Let's assume that we are playing a popular version of video poker which pays on any pair of Jacks or Better and does not use any wild cards.  This version (known as 9-6 Jacks or Better) offers the pay schedule shown in Table 2.

This version of video poker is one of the best around for both long-term and short-term play and is found throughout Nevada.  The "9-6" refers to the payoffs for Full Houses and Flushes.  Casinos commonly "monkey around" with these payoffs.  Thus you will find 8-5 and 6-5 versions of the game, where the payoffs on a Full House have been reduced from 9 for one to 8 for one or 6 for one, and the payoffs on the flush reduced from 6 for one to 5 for one. 

These may not seem like big reductions in payoffs, but they make a huge difference in how beatable the game is.  We will go over the different versions of video poker in a couple of chapters, but for now, let's just assume that we have found a 9-6 Jacks or Better video poker machine and that it has the following payback schedule:

Table 2.  Pay Schedule for 9-6 Jacks or Better

Any winning poker hand in this version of video poker will pay off in accordance with this pay schedule.  If we are dealt a high pair, say a pair of Kings, then our payoff will equal the amount of money wagered.  Any other winning hand will be paid off in the same manner.  If we have a straight, we will get 4 times our wager, a flush will pay 6 times our wager, and so on.

In the case of winning hands, the value of the hand is simply the amount shown on the machine's pay schedule.  To simplify matters, we will assume that the amount wagered is one dollar and express all values in dollars.  Using this approach to valuing hands, a straight is worth $4 and a flush $6.

To obtain these values, we are really multiplying our possibility of winning times and potential payoff.  With made hands, our possibility of winning is certain, that is 100%, which is expressed mathematically as 1.0.  To determine the value of a hand, we multiply the probability of making the hand times the payoff for making the hand.  Thus the value of a made flush, such as 2©4©7©8©J©, is 1.0 x 6 for a value of 6, which we will call $6.00.

With hands that are not yet winners, we can use the same approach to evaluate them.  We can multiple the probability of winning with that hand, times the payoff if the hand wins.

This approach to evaluating the value of different poker hands is called calculating the Expected Value of the hand. In calculating an expected value, we have a simple way of comparing the value of one poker option with another.

Going back to our hand of 5©6©7©8§8¨, we could evaluate all the possibilities of keeping the pair of eights and drawing three cards by looking at all of the possible combinations of hands.  There are 16,215 possible draws, which would include 4 of a Kind - 45 times, a Full House - 165 times, 3 of a Kind - 1,854 times, Two Pairs - 2,592 times and no value hands - 11,559 times. 

To convert this information into a form we can use for computing the value of different options, we must multiply the frequency of each hand times its possible payoff and compare these values with the total number of draws.  Table 3 shows these calculations for discarding three cards and drawing to a low pair.                           

                 Table 3.  Expected Value of Drawing to a Low Pair

 
Before your eyes start to glaze, relax.  I am not going to make you do any calculations like this.  I just wanted to show you what's involved in computing the expected value of a hand.

This means that the value of keeping the low pair and drawing three cards is $.82.  Anything less than a dollar means that our whole bet won't be returned.  So this hand is going to be a loser on the average.  But that doesn't mean that we shouldn't play the hand to its highest potential.  Let's take a look at the other options of drawing to a 4-card straight or a 3-card straight flush.

Computing the expected values for these hands as well as the low pair, we have:

Hand                                                                            Expected Value

Keep Low Pair                                                                              $.82

Keep 4 Card Straight                                                                      .68

Keep 3 Card Straight                                                                      .58

When faced with a decision like this, calculating the expected value makes our decision of what to do easy.  Our basic rule of play is to always go for the hand with the highest expected value.  In this case, we will keep the pair of eights and draw three cards.

Let's consider another hand of T©J©A©2©2¨.  If we hold just the low pair of twos, the value of this hand is $.82.  But what if we decide to go for the Royal Flush and hold the T©J©A© and draw two cards?  The expected value of this option in the 9-6 Jacks or Better version of video poker is $1.32.  But there's still yet another option isn't there? 

Let's see what happens if we hold four Hearts and go for a flush.  The value of this option is $1.28.

            Here's a summary of our three options:

1.  Hold the low pair of Twos                                                        $.82

2.  Hold the 10, Jack and Ace of Hearts                                        1.32

3.  Hold the four Hearts                                                                 1.28

The calculations show us that the prudent course here is to keep the T©J©A© combination, and discard the low pair.  You may think that it is the possibility of making a royal flush that makes this option more viable, but the royal flush can only be made one way with this draw. 

There are 1,081 combinations of two cards that can replace the pair of Twos.   These include a pair of Jacks or Better - 240 times; Two Pairs - 27 times; Three of a Kind - 9 times; a Straight - 15 times; a Flush - 27 times and only one way of making a Royal Flush.   Adding up the values for each of these possibilities gives us a value for the hand of $1.32 and tells us that discarding the low pair is our best option.

Your approach to playing video poker hands should now be obvious.  You should always play the hand with the highest rank or value. 

The above information was taken from the Power Video Poker manual.

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