Ask the Slot Expert: I have to watch what I write

When I'm writing this column, I sometimes find myself about to type something only to have to rephrase the point I want to make because my original phrasing was not mathematically accurate.

When I was an expert witness on a slot-related case a number of years ago, the attorney I worked with told me that they had read many of my columns and couldn't find any contradictory statements in them.

The attorney asked me to sit in on the deposition of the other side's expert witness. During the questioning, my attorney pointed out a few contradictory statements in this expert's columns. The expert said that his primary goal was to be entertaining, not to be a math teacher.

I try to make this column entertaining and mathematically accurate. I can control the latter; you're the judge on the former.

I sometimes use the tossing of a fair coin to illustrate random concepts. Let's say we toss it 10 times and get 8 heads. Not exactly 50/50, but this is the short run. What do we anticipate to see if we keep tossing the coin?

You might be tempted to say that the number of heads and the number of tails will get closer to being equal as we continue tossing. You might also say that the number of heads will get closer to half the number of tosses, as will the number of tails. These statements are all saying the same thing: number of heads = number of tails = half the number of tosses.

They're also all wrong. I once was about to write that the number of heads would equal the number of tails, but that's not what tends to happen.

I had to take a breath and make my statement accurate. What does tend to happen as we continue tossing the coin? The ratio of heads over tails tends to get closer and closer to 1.

I found a coin toss applet online to save me the trouble of tossing a coin. The following table shows the number of heads and tails I had after a certain number of tosses, the absolute value of the difference between the two numbers, and the ratio of the two numbers.

If the number of heads and the number of tails got closer together the more we tossed, the fourth column would get closer to 0. It doesn't. In fact, it tends to increase. Nevertheless, the ratio of heads to tails, the fifth column, does however tend to get very close to 1.

To quote from The Case of the Gambling Nobleman chapter in Conned Again, Watson by Colin Bruce:

...in a long run of coin tosses, it is only the ratio of heads to tails that tends towards unity, and the absolute discrepancy between the number of heads and tails could only tend to increase.

Note how Bruce and I always say "tend." Because we're dealing with a random event, unlikely events sometimes happen. The best we can do is say what will tend to happen. There may be exceptions.

You can see in the table the discrepancy between heads and tails tends to increase, but it sometimes decreases. The ratio tends to get closer to 1, but sometimes it falls a little below 1.

I almost got tripped up writing this column. A few paragraphs ago, I had originally written "What do we expect to see....", but I had to go back and change "expect" to "anticipate". "Expect" has a mathematical meaning (as in Expected Value) that I did not want to imply.

If you'd like read more about why heads does not equal tails and why I didn't use the word "expect", see this question on Quora.com: I suspect that the imbalance between heads and tails is likely to increase the more I flip a coin.

It may be clunky prose to write about ratios and it may seem wimpy to write what tends to happen, but I have to write this way to try to be mathematically accurate.

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