Calculations aren't always what they show

March 25, 2008 6:00 PM
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Winning Strategies by Elliot Frome |

Last week, I showed how by some relatively simple manipulation of numbers, a person can honestly claim to beat the house 80 percent of the time.

It is quite interesting that while math is an absolute, it does not mean that you can’t manipulate the words around the numbers to greatly change their meaning. Let’s look at a relatively simple example within the casino.

In craps, the ‘hardways’ wager looks for both dice to have the same value. So, when you wager on the hardways 10, you are looking for Double 5’s. The wager will lose if the roll is equal to any other 10 (easy way 10) or any 7. The other rolls are all meaningless to the wager.

So, what is the payback for this wager? There is one way to win (5-5), and eight to lose (4-6, 6-4 and the six ways to make a 7). The other 27 rolls are meaningless.

The wager pays 7-1, so essentially the player will have returned to him eight units for every nine wagered for a payback of 88.89 percent. I have to admit, I personally find this number to be rather misleading.

There are, in effect, 27 rolls for which this wager will push. Further, casino rules allow you to pull down the wager after one of these ‘pushes’. So, why do we not account for these rolls in our payback calculation? If we were to do that, the calculation comes out a bit different. We would receive an additional 27 units back (for a total of 35) out of every 36 rolls.

Now the payback is 97.22 percent, which sounds a bit more respectable.

I can’t really give a good reason why this wager’s payback is not calculated in this manner. It would be one thing if it were common to ignore pushes when computing paybacks, but this is not the case.

Further, since you are allowed to pull back the wager after each push, it can be considered as if each roll is an independent wager. This is unlike the pass line wager, which also can last for many rolls, but must stay out there until it is resolved to a win or lose finish.

This made me start thinking about other games and the potential impact if we ignored pushes. In the case of our hardways wager, pushes account for 75 percent of the possible outcomes, and the impact is rather large. The impact will be larger as the frequency of pushes goes up because pushes count as 100 percent payback. This weighted average will have a larger impact.

Similarly, the lower the payback is without pushes the more the pushes will push the payback up.

For example, in Three Card Poker, pushes are relatively rare – accounting for only about 0.1 percent of our hands. With a payback of about 98 percent for ante/play, if we were to exclude ties, we’d barely notice the difference in our overall payback. In blackjack, ties make up about 8 percent of the hands. However, because the payback is already so close to 100 percent (at about 99.4 depending on the exact rules), the impact is still roughly a miniscule 0.05.

Again, this is barely noticeable.

But, what about a game like Pai Gow poker, where pushes occur more than 41 percent of the time. Including these pushes in the payback calculation yields a payback of 97.27 percent (when the player is not the banker). If we ignore the pushes, we find a rather different situation. In this case, the payback drops to 95.33 percent.

What about video poker? The first reaction for many of you will probably be pushes in video poker? Absolutely! When you get a pair of Jacks or better, all you are doing is pushing. The machine pays 1 for every 1 unit wagered. This occurs about 21-22 percent of the time in video poker.

 If we were to treat these hands like we do the hardways rolls that have no meaning, the payback of full-pay Jacks or Better would go down by about 0.1 percent. This is not as dramatic as the example with Pai Gow poker, but it would increase the house advantage by 20 percent.

But, as I stated earlier, my goal is not to ignore these pushes, but rather the opposite. I think the more proper payback for the hardways 10 wager is really 97.22 percent. The average pass line wager is out there for probably 3-4 rolls.

Even though the player can’t take them down once the point is made, I still think the amount of rolls that the wager is out there needs to be taken into account in determining the payback. That way a player can fairly compare paybacks.

If the casino created a new side bet whereby the player were paid based on how many 7’s come up in 10 rolls of the dice, would it be fair to compare this to a wager that pays on how many 7’s occur in 100?

My point is that you can’t let someone distort the math by distorting the words around the math.