When I explain how an Expected Value is calculated in an article, as I did last week, I find that it leaves many people still a bit confused.
Thus, it is often useful to provide an example or two that shows exactly how the concept works and why. So, I figured this week, I’d walk through a couple of video poker examples.
Let’s start with a very simple initial deal, such as: 3d, 3c, 3h, 8s, Qh
You don’t have to read my books, my columns or anybody else’s to know how to play this hand. You’re going to keep the three of a kind. However, just because it is this simple does not mean that essentially your brain made this decision after analyzing the other 31 possible ways to discard. It just did it very quickly, and realized that you’re going to win WAY more money by holding onto the 3s.
Even though this is an obvious hand, it doesn’t mean that it doesn’t have an Expected Value. We calculate this value by playing out every possible draw once we hold only the three of a kind. There are 1081 possible 2-card draws from the remaining deck. Of these, 46 will be four of a kind, 66 will be full houses and the remaining 969 will be three of a kinds.
We multiply these values by the payouts of these hands (25, 9 and 3 respectively) and add them up to arrive at a value of 4,651. We then divide this by the 1,081 possible hands to arrive at our expected value of 4.30. If we were to do the same thing for any of the other possible ways to play the hand, it wouldn’t even come close to this value, which is why this is the best way to play the hand.
If only the hands were always that easy. Of course, if they were all that simple, I’d be out of a job. So, let’s look at a more complex case. What if you are dealt:
5d, 6d, 7d, 8s, 8c
Of the 32 ways to play the hand, 29 will be quickly discarded, leaving 3 possible plays. You can either hold the 3-Card Straight Flush, the 4-Card Straight or the Pair. In each case, we are drawing a different number of cards, which is why we divide our result by the number of possible draws.
We want to make sure we compare apples to apples. So, a computer program plays out each of the possible draws for each of our three scenarios and then sums up the payouts of each paying hand. When this is done, we find that the 3-Card Straight Flush has an expected value of about 0.63. The 4-Card Straight has an expected value of about 0.68, and the Low Pair has an expected value of 0.82.
Expert Strategy dictates that we play the way that has the highest expected value. So, the proper play is to hold the Low Pair. It should be noted that because the expected value is below 1.00, it means that in the long run this hand is a loser.
Very few hands are actually winners in the long run. So, in this case we are actually trying to minimize our loss. It is not necessary to memorize these expected values, but you do have to memorize the relative order of each hand in the strategy table. From this example, we learn that we play a Low Pair over a 4-Card Straight with no High Cards and a 3-Card Straight Flush with no High Cards. We also learn that if we didn’t have the Low Pair, we would play the 4-Card Straight over the 3-Card Straight Flush.
You can try out your strategy by playing our video poker game.