Ask the Slot Expert: I have to watch what I write - Part 2

Question: I just wanted to comment on your column about accuracy and being specific about your terminology.

First of all, on the subject of being entertaining being judged by we, the readers. Obviously I vote yes, otherwise I wouldn't be commenting.

As far as the terminology goes, I think most of your regular readers understand the differences, but many others probably don't even understand the differences.

Another point would be that SOME of the regular readers (not me) would likely (possibly?) call you out if you did, for example, use expect instead of anticipated, etc. So it probably is a good idea to make sure there is no ambiguity in the terms you use.

But actually, the main reason I am sending a comment is to point out something funny in the applet example you use. The ratio column shows a crossover of heads to tails from under 1 to over one, which means that at some point the number of heads and tails actually were equal.

I know that it isn't necessary that this happens, but the comment that the number of each actually get farther apart isn't any more likely than getting closer. They COULD get farther apart while the ratios get closer to 1, but they could also get closer.

Thanks for your columns and thanks for reading my comments.

Answer: Thanks for the kind words.

Last week I used the example of getting 8 heads out of 10 tosses of a fair coin and wondered what we think will happen if we kept tossing the coin. I had originally used the verb "expect" (What do we expect....) but went back and used "anticipate" (What do we anticipate....) instead. "Expect" has a mathematical meaning that I did not want. I wanted what our gut tells us should happen.

I'll try to use "anticipate" for what our gut feeling and "expect" for what the math tells us.

Last week I said that if you wanted to read more about why the number of heads tends to not equal the number of tails and why I didn't use the word "expect", you could see this post on Quora.com: I suspect that the imbalance between heads and tails is likely to increase the more I flip a coin.

Did you read that post? No? I don't blame you.

I'll quote the relevant paragraph from Dr. Amit's answer here.

After flipping the coin n times, your distance from the origin is expected to be around SQRT(2n/p). (This magnitude is the expectation of the distance for large n.) It could be more, or less, but on average that’s what it’s going to be. So after flipping a coin 100 times you expect to have a discrepancy of about 8 between the number of Heads and Tails, but if you patiently flip a million times, you’d expect a discrepancy of 800.

After a large number of tosses, we anticipate (gut feeling) being on the origin, heads=tails, but we expect (mathematically) to tend to be farther away. Anticipation is our fantasy; expectation is mathematical reality. The difference between anticipation and expectation built the mega-resorts on the Las Vegas Strip.

Dr. Amit points out that "expect" deals with averages. Last week, I also said that I use the verb "tend" frequently. Because we're dealing with random events, I can only say what will tend to happen. As we toss the coin, we tend to get farther away from heads=tails, but sometimes we get closer and sometimes we have more heads than tails and sometimes we have more tails than heads. But if we did a bazillion sets of n coin tosses, the average value of the absolute value of the difference between heads and tails is given by that formula that has pi in it. We expect to be that far away from heads=tails. (And speaking of pie, a slice of the apple pie I got from Red Rock this past Thanksgiving sounds good right now. The pie is okay. It's still frozen.)

Why pi? I don't see any circles in the coin toss -- except for the coin itself. The document Simple Random Walk from University of Chicago Math shows how pi is used to approximate the value of large factorials. Be forewarned that the only thing simple about this document is the random walk. The math is not so simple.

If we're going to have an equal number of heads and tails, we know we have to have an even number of tosses. To guarantee an even number of tosses, let's say that there are 2n tosses. Out of 2n tosses, we have n heads. The number of ways we can distribute n heads throughout 2n tosses is the combination of 2n things taken n at a time, which is (2n)!/(n!n!). If n is large, the factorial is large and it's time for pi(e). (I like my apple pie with a slice of cheddar melted on top.)

Here's the chart I printed last week. I changed the fourth column to show just tails-heads instead of the absolute value of tails-heads.

As you pointed out, we had an equal number of heads and tails whenever the ratio in the fifth column went from under 1 to over 1, and vice versa. That was very astute. I don't know how many people realized that happened. To make it easier to see when we had heads=tails, I changed column four to show the actual difference between tails and heads. One of those events happened whenever the sign of the value in the fourth column changed.

Actually, there could have been many more instances in which heads=tails. My chart only shows where we were at the end of the interval. It doesn't show what might have happened during the tossing.

If you would like to see more non-smoking areas on slot floors in Las Vegas, please sign my petition on change.org.