Today’s paper is on simple video poker mathematics. Let’s assume you are playing a game where, on average, you hit a quad (i.e., a 4-of-a-kind) every 400 hands. Further, let’s assume you play for a total of 1,200 hands. I’ll arbitrarily say that it takes you two hours to complete the 1,200 hands. How many quads can you expect to end up with over that number of hands?
It appears obvious that the answer should be three, but this is the wrong answer. To get the correct answer, we need to look at the binomial distribution, the results of which appear here:
| 0 | 5% |
| 1 | 15% |
| 2 | 22% |
| 3 | 22% |
| 4 | 17% |
| 5 | 10% |
| 6 | 5% |
| 7 | 2% |
| 8 or more | 1% |
What this says is that 5% of the time you won’t hit any quad; 17% of the time you’ll hit four; 2% of the time you’ll hit seven; etc. These numbers don’t tell you WHICH quad you’ll hit. Just how many.
These numbers are accurate, but not really precise. For example, the chance to get exactly three quads could more precisely be written as 22.4322%, but that is far more precision than we need for today’s discussion. It looks like they only add up to 99%, but that’s rounding error and also not important for today.
One of the interesting features of this distribution is that the number of quads that we think we “should” get, namely three, actually occurs less than one time in four. Another typical feature of the distribution is that the probability of getting one fewer quad than typical is virtually the same — actually 22.4135%, which is slightly less.
We could, I suppose, refer to getting either zero or one quad as “bad luck”, getting two, three, or four as “typical luck”, and getting five or more as “good luck”. It doesn’t change anything by assigning terms dealing with luck to the results. When somebody asks me, “How much skill and how much luck was involved?” in describing whatever happened yesterday, my answer is often, “I have no idea.”
Let’s assume that on this particular day in question, we don’t hit any 4-of-a-kind. Definitely worse-than-average luck, but it happens about one day in twenty. Slightly rare, but not extraordinarily so. Now the question is, since you’ve just gone through worse-than-average luck, what will be the distribution of quads for your two-hour session tomorrow? For this, the following distribution will hold:
| 0 | 5% |
| 1 | 15% |
| 2 | 22% |
| 3 | 22% |
| 4 | 17% |
| 5 | 10% |
| 6 | 5% |
| 7 | 2% |
| 8 or more | 1% |
The distribution, of course, is the same as first given. Just because we had a bad day says absolutely nothing about what our score will be the next day. There is no tendency to either, “Once you start running bad you keep running bad because you’re an unlucky player,” or “You’ll get more quads the next day to make up for the shortfall.”
Let’s assume we change machines halfway through. Now the distribution of the quads expected over the 1,200 hands is:
| 0 | 5% |
| 1 | 15% |
| 2 | 22% |
| 3 | 22% |
| 4 | 17% |
| 5 | 10% |
| 6 | 5% |
| 7 | 2% |
| 8 or more | 1% |
Is this distribution beginning to look familiar? It should. Changing machines has nothing to do with changing the distribution.
In this discussion so far, we’ve said nothing about skill. We are assuming players are playing perfectly. If players play imperfectly, the distribution will change. For example, on a hand like K♥ K♠ 4♦ 4♣ 5♦, it is correct in almost every game to hold KK44, although many seat-of-the-pants players playing games where two pair only return even money incorrectly hold just the pair of kings. Making this kind of mistake systematically will IMPROVE your chances for hitting quads, but COST you overall. The increased number of quads you get by holding only one pair rarely compensates for the reduced number of full houses.
The numbers are for three “cycles.” If full houses normally come around every 90 hands on average, the numbers above apply to how many full houses you hit in 270 hands. If royals come about every 40,000 hands, the numbers above apply to how many royals you hit in 120,000 hands. In games where the royal cycle is 45,000 hands, the numbers apply to how many royals you hit in 135,000 hands.