Dice rolls help to explain actual probability

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For those who have read my column through the years, you know I stay apolitical. So, what I’m about to write is not a political commentary in any way but rather a simple example of how you can make numbers say just about anything you want.

If I were to ask someone whether we have generally had a Democrat or a Republican as President, what would the correct answer be? My guess says that it will mostly depend on what message the person responding is trying to present.

After all, we’ve had a Democrat for the past 4 years. Or have we had a Democrat for 4 of the past 12 years? Or have we had a Democrat for 12 of the past 20 years? Or have we had a Democrat for 12 of the past 32 years? 

So, depending on your point of view, we may have had a Democrat as President for 100% of the past few years, 33% of the past few years, 60% of the past few years or 37.5% of the past few years. I could just as easily put these in terms of % of Republican years and the numbers would be all over the place as well.

On the one hand, math is an exact science. On the other, it can be so easily distorted to say whatever the messenger wants it to say. Throw in the fact that most people don’t fully comprehend all the nuances of things like probability and it is very easy to be misled by those who wish to do so.

Last week, my son, who is in kindergarten brought home a project he did in school. They rolled a single die multiple times and recorded the results. Check the chart for the results that were tallied (the exercise was for this purpose – tallying)

When I first saw this, my initial reaction was “looks like a loaded die”. Of course, I said this tongue in cheek. But, it did start to make me wonder about how someone else might react and what the actual probabilities are for each of these occurrence counts. Obviously, we would expect an average of 5.5 rolls for each number. How outrageous was this result? Could it have been a loaded die?

Well, I can’t speak to the fairness of the die. Barring any proof to the contrary I have to assume that it was a fair die or at least as fair as any die you might get in a board game. I doubt it was approved by GLI or NGCB, but I’m going with the notion that it was fair.

To begin with, let’s dispel what might be one misconception about this project. Just because the average is 5.5 does NOT mean the probability of a 5 is the same as that of a 6. They are close, but the probability of 5 is 18.5% and the probability of 6 is 17.3%.

So, what is the probability of a number coming up 8 times? An astronomical 8.7%. In other words, it is rather common, albeit not the most frequent occurrence. It would take a single number coming up 15 times before we hit something that was more rare than 1 in 10,000.

And, as I said in the beginning we can all make numbers say what we want with a little slight of hand. The numbers presented here are dependent on one another. In 33 rolls, if one number shows up 8 times, there are only 25 more rolls for the other 5 numbers. That is NOT the same as I rolled a die 33 times and just counted how many 1’s I got and then repeated the entire thing multiple times. The results in this case would vary from the example I presented.

So, what is the point of all this? Pay careful attention when someone just throws a bunch of numbers at you. You should make sure that you are comparing apples to apples.

Numbers can be spun and it usually takes numerous pieces of key information to get a full picture of what is going on. Most importantly, very little of what actually occurs is “outside the norm,” meaning games are not rigged.

The last leg of Expert Strategy is knowing what to expect and this is by far the largest and in many cases the most important aspect of the strategy. In the case of our little die, rolling 1 eight times was very normal. But, if you were playing a game based on it and began to convince yourself the die is rigged, you might find yourself believing that a 1 is truly more likely than the other numbers and then change your strategy based on it.

The casinos are hoping this is what you will do.

Elliot Frome is a second generation gaming analyst and author, whose math credits include Ultimate Texas Hold’em, Mississippi Stud and many other games. His website is www.gambatria.com. Contact Elliot at [email protected]


About the Author

Elliot Frome

Elliot Frome’s roots run deep into gaming theory and analysis. His father, Lenny, was a pioneer in developing video poker strategy in the 1980s and is credited with raising its popularity to dizzying heights. Elliot is a second generation gaming author and analyst with nearly 20 years of programming experience.

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