Wednesday, December 21, 2011

Clarity on a Little Advice

Last week’s column gave some simplistic advice to beginners who are not yet ready to sit down and really learn the strategy for video poker. It discussed the relative rankings of four of the most common hands – high pair, four-card flush, low pair and four-card straight.
While I gave the expected values for each of these hands, along with some explanations as to why the rankings are what they are, this week I want to stress that these explanations are not the critical part of the process.
The strategy for video poker is based on one thing – math.
We don’t keep a high pair over a 4-card flush because the high pair is a sure winner. If this were the case, we’d keep a high pair over a four-card straight flush, too (but we don’t!). The fact that the high pair is a sure winner explains why its expected value is as strong as it is, but it is the actual value of this expected value that puts the high pair where it does.
So what is this "expected value" I keep talking about?
It is the average amount of coins we expect to win over the long run from that hand.
How is it calculated?
It is calculated by looking at EVERY possible draw given the 5-cards already dealt.
Say what?
There are 2,598,960 ways to deal five cards from a 52-card deck. For each of these ways, there are 32 different ways to play each – ranging from discarding all the cards to discarding none of them. For each of these 32 ways to play a hand, there is a varying number of possible draws.
If we discard one card, then there are 47 possible draws (each of the 47 remaining cards). If we discard three cards, then there are 16,215 possible draws (choosing three cards from 47). A computer program goes through every possible draw and tallies up the winning hands for each of the 32 ways to play a hand.
It then computes the average number of coins returned for that way. This is the expected value for that particular way of drawing. It compares the expected values for each of the 32 ways and whichever has the highest one is the proper play for that deal and is deemed the expected value for that deal.
An example usually helps to shed some light on this process. Assume you are dealt: 4 of clubs, 5 of hearts, 5 of clubs, 5 of spades, 7 of diamonds.
We recognize the three-of-a-kind (5’s), the EV of which is calculated as follows:
Drawing two cards from the 47 remaining in the deck will create 46 four-of-a-kind winners (a five combined with each of 46 remaining cards). Sixty-six draws will end as full houses (six pairs in all ranks but 4, 5, and 7; 3 pairs of 4 and 7) while the remaining 969 draws do not improve the hand but instead leave it as a three-of-a-kind.
In summary we have:
46 4-of-a-Kind paying 25 each,
66 Full Houses paying 9 each,
969 3-of-a-Kind paying 3 each,
We calculate the total payout as 4,651, which is an average of 4.30 for each of the 1,081 possible draws. Therefore, the expected value of this deal/draw combination is 4.30.
As should be fairly obvious, if we try to play this hand in any of the other 31 ways, the expected value will NOT be any higher than 4.30 and thus this is also the expected value of this deal.
As all three-of-a-kinds have exactly the same expected value, this is ALSO the expected value of all. We will find this value on our strategy table.
Next week, I’ll walk through the four hands (high pair, low pair, four-card flush and four-card straight) I used in last week’s column. This will explain why the strategy I described last week doesn’t just make some sort of logical sense but is the right play mathematically.
I’d like to take this opportunity to wish everyone a happy holiday and a very happy and healthy 2012!