Over the past few weeks, I have described how the expected value is calculated for video poker. While the exact methods vary somewhat from game to game, the concept is still the same.
This is the most critical aspect of expected value. It is the expected return of the hand over the long run. I think it should be clear that it cannot be considered to be some sort of expected return that you will earn on this specific hand.
For example, the expected value of a High Pair in jacks or better video poker is 1.54. But the only possibilities are that you’ll get back 1, 2, 3, 9 or 25 units depending on whether you wind up with a High Pair, Two Pair, Three of a Kind, Full House or Four of a Kind.
The greater the number of High Pairs you have the more likely that the overall result of all these hands will approach that 1.54. If we were to think about it for a minute, we’d realize that since we will earn a minimum of 1 and a maximum of 25, but the average is just over 1.5 — most of our hands will wind up as High Pairs.
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Another key point about expected values in video poker to keep in mind is that the expected value in the strategy table is the average expected value of all hands of that particular type. In the case of a High Pair, all High Pairs have the same expected value. The three cards that are discarded should always be three cards of different ranks. If you are holding a Pair, there is no way to get a Straight or a Flush. So the probability of winding up with something other than a High Pair is the same for every High Pair.
But the same is not true for 4-Card Flushes. I described last week how Straights are broken down by the number of High Cards, but we don’t do that for 4-Card Flushes. That doesn’t mean that Flushes with more High Cards have the same expected value as one with less High Cards. We are able to lump them together on our strategy table because the number of High Cards does not change the strategy the way it does for 4-Card Straights.
But just like partial Straights, the more High Cards, the higher the expected value. The 1.22 we see listed on the strategy table is the average of all 4-Card Flushes. Ironically, there are no 4-Card Flushes that actually have this expected value. They range from about 1.10 to about 1.35. It would go even higher, but if you have a 4-Card Flush with 3 High Cards, this becomes a 3-Card Royal with the expected value of 1.41.
Again, not all 3-Card Royals are the same either. A J-Q-K 3-Card Royal has a higher expected value than a 10-J-K 3-Card Royal. Not only do we have 3 extra cards to potentially pick up a High Card, but there are also more ways to turn a Straight with the J-Q-K than there is with a 10-J-K.
Additionally, it is possible that for either one we are discarding a card which made the hand a 4-Card Straight or a 4-Card Flush. If this is the case, then we are reducing the expected value because we discarded a card that we would otherwise want as a draw.
Quite frankly, this is mostly done because of the power of the payout of the Royal Flush. If we hold that 4th card to make a Straight or 4-Card Flush, we will give up any chance to get the Royal. But with an 800 payout (assuming max coins), we are willing to reduce our chances of getting a Straight or a Flush to give us a chance to get the Royal.
Of course, there are still may opportunities to draw to a Straight or a Flush, but there will be even more of them if we discarded a 2 and 5 of a different suit rather than a card that could have potentially helped us.