# Long-run behavior in coin tossing experiment

I have read the following information on this site and I am not sure if it is actually true (or if I misinterpret it):

Events that are random are not perfectly predictable, but they have long-term regularities that we can describe and quantify using probability. In contrast, haphazard events do not necessarily have long-term regularities. Moreover, we limit the term probability to events for which we can specify all possible outcomes. How a fair coin lands when it is tossed vigorously is a canonical example of a random event. One cannot predict perfectly whether the coin will land heads or tails; however, in repeated tosses, the fraction of times the coin lands heads will tend to settle down to a limit of 50%. The outcome of an individual toss is not perfectly predictable, but the long-run average behavior is predictable. Thus it is reasonable to consider the outcome of tossing a fair coin to be random.

Let's assume 100,000 coin tossing trials in total and a perfectly fair coin. If we observed that in the first 1000 trials, Heads occurred 600 times and Tails 400 times, then what the above paragraph tells us is (the way I understand this): we could predict that at some point, in the next 99,000 trials (and maybe for a relatively short number of trials), Tails will start coming more often than the Heads so that the final result (of 100,000 tossings) will be approximately 50,000 Heads and 50,000 Tails. Is that interpretation correct or is such predictability rule valid? If yes, is there a way to show this by Bayesian methods?

(This is not a student exercise. I am a postdoc in Bioinformatics and, probably, a bit confused.)

• Another problem with the passage you quote is that there is no clear definitions for the terms "random" and "haphazard". Jan 23, 2014 at 20:16

After 1,000 tosses you have 600 heads and 400 tails. What the fairness of the coin tells you is that in the next 99,000 tosses you are likely to get about 44,500 heads and 44,500 tails. But that would mean that, in the whole 100,000 tosses you would have 45,100 heads and 44,900 tails, which is very close to 50%.

Your interpretation is known as the "Gambler's fallacy".

In the passage, the phrase "entirely predictable" is misleading; a single toss of a fair coin is not predictable at all (except that you can say "it will be heads or tails".

In essence, no. The trials are independent and identically distributed (IID), so the probability of heads in the next turn is always 50%. If you think about it... Does your coin know that you're going to throw it 100000 times?

• That's right, I would add IID though :-) Jan 23, 2014 at 10:05