Ask the Slot Expert: I have to watch what I write - Part 3

Trying to be mathematically accurate in my columns leads to consistency. I realized this last night watching an old episode of History's Greatest Mysteries (HGM). This episode was about the pyramids on the Giza plateau in Egypt.

We all know there are three large pyramids in the complex: one built for Khufu, one for Khafre, and one for Menkaure. Father, son and grandson -- nepo babies. One pyramid is known as the Great Pyramid because it is larger than the other two. One pyramid looks like the tippy top of it has been lopped off. One pyramid still has some of the limestone casing at its apex.

Do you know which is which?

HGM correctly identified the Great Pyramid in its initial description of the Giza complex, but then showed the wrong pyramid while discussing the Great Pyramid in some later scenes. I've seen many other programs talk about the Great Pyramid and then show a picture of a not-Great Pyramid.

I wonder whether programs show the wrong pyramid because the Great Pyramid isn't the sexiest pyramid of the bunch. The Great Pyramid isn't the pyramid that still has some of the casing stones at its peak and looks like a complete pyramid. The Great Pyramid is the one that is missing its peak. It is not the pyramid in the middle of the three.

The Great Pyramid was built for Khufu. It is "great" because it is the largest of the three pyramids. The length of its sides are about 775 feet and its original height was 481 feet. Khafre's pyramid, the sexy one in the middle, is 707 feet on its sides and 471 feet tall. Menkaure's pyramid is only 365 feet on its sides and 218 feet tall. (The ancient Egyptians were getting kind of tired of the whole pyramid shtick by the time he came into power.)

Khafre's pyramid appears to be larger in long shots of the trio because its site is slightly higher than that of the Great Pyramid and it has a steeper angle of inclination.

Khafre's pyramid looks like what we think a pyramid should look like, so some programs sacrifice accuracy for entertainment value.

I have to be careful in explaining why your run of extraordinarily good (or bad) luck has little effect on a machine's long-term payback. Some people may be tempted to say that one run would be offset by an opposite run, so in the end the machine "makes its numbers." But that's not what tends to happen.

As we saw the past two weeks in the coin flip example, we actually tend to get farther away from heads=tails when we toss the coin a large number of times. (I've found a coin-flip simulator that can perform one million flips per run. I'll execute many runs and see how close we get to the average of heads-tails given by the formula SQRT(2*n/pi), 800 for 1,000,000 flips.) Nevertheless, the ratio of heads over tosses (and tails over tosses) tends to get closer to 0.5.

Similarly, the amount of money returned by a machine over the total amount of money played in a machine tends to get closer to the machine's long-term payback as the machine gets more play. Extraordinary runs are a smaller and smaller piece of a machine's total performance as players keep playing the machine.

It's easier to visualize this effect using a coin toss and the Drunkard's Walk.

You and a friend are standing in a flat, barren desert. You're both facing north and your friend is directly in front of you. He takes a step forward and tosses a coin. If it's heads, he takes a step to the left. Tails, a step to the right. Then he takes another step forward and repeats the coin toss and step to the left or right. The farther he walks away, the closer he will appear to being exactly north of you.

This is because the ratio between his distance north of you and the distance he has veered to east or west is likely to diminish. So by the time he is but a dot on the horizon, you will see him very close to due north.

Conned Again, Watson by Colin Bruce

Another quote from the book that illustrates the point.

Any advantage of heads over tails, or of red over black on the roulette wheel, is smoothed out in the longer term, not by any kind of active cancellation, but by a more gentle and undirected happening, a kind of washing out. That is, the effect of any flukish run gradually becomes ever more negligible relative to the whole.

The example I use is comparing a drop of food coloring in a glass of water and a drop of food coloring in the Atlantic Ocean. Your experience is the glass of water. The drop has a big effect on the color of the water in your glass. That same drop has almost no effect on the color of the Atlantic Ocean.

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