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Interpreting House Edge, Part II2 May 2009
Last month we observed that one must be careful when interpreting the house edge. This month the lesson is that one must take care when calculating it.
A common wager at Craps is to place both the six and eight. When placing the six, for example, one is wagering that the shooter will roll a six before rolling a seven. If the seven occurs first the bettor loses. The payoff for a winning wager is 7:6. It is easy to calculate the house edge for these bets. The six, for example, is settled by only two numbers, the six and the seven. There are five ways to roll a six and six ways to roll a seven so the table looks like the following.
So if you wager six units your average loss will be 1/11 units. The house edge is a figure based on a per unit wagered basis, so we have to divide by six. Thus the house edge as a fraction is 1/66 or approximately 1.5152%.
Last month I showed you that if p is the probability of success in a series of independent trials then 1/p is the expected number of steps to a success. If we deem resolution of the placed six as a success, then the average number of rolls to a resolution will be 36/11. Using the figure of 108 rolls per hour (the figure I used last month) there will be 11/36 x 108 or 33 resolutions per hour on average. For a six-unit wager the average loss will be 1/11 x 33 or three units per hour. The placed eight will be identical so our average loss placing six units on the six and eight will be six units per hour.
Now let me show you how not to analyze this wager. If either the six or eight occur the player wins seven units and if the seven occurs the player loses twelve units. So we construct the following table.
The house edge will be 1/8 divided by the total at risk, which is twelve, so we obtain 1/96 or approximately 1.04167%. It would appear that by placing both the six and eight we have lowered the house edge. What's going on?
The problem here is that if (say) the six is rolled the eight has not been resolved but the above table treats it as if it were. It would be a phony game in which the six is paid, the eight is taken down, and then put right back up again (which would not endear you to the dealer). But you ask, can't you just pretend to take the eight down and put it back up? Sure, but it is still a phony game and the phony house edge hasn't helped your resources one bit. Let me show you why.
In this game the probability of a resolution is 16/36 or 4/9. Hence there are 9/4 rolls per resolution. Using 108 rolls per hour we would have 4/9 x 108 or 48 resolutions per hour. From the above table each resolution pays 1/8 unit, so the average loss per hour would be 1/8 x 48 or six units. This is the figure we calculated earlier.
You see, we invented a game that appears to lower the house edge. However it also increases the number of resolutions per hour so the net result is that you will still lose six units per hour. This same kind of fuzzy reasoning was addressed in my May 2003 article entitled Mensa Mystery. You can find it in the archives and you might find it interesting to go back and read it. See you next month.
Don Catlin can be reached at email@example.com
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