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I have a question on the fundamental meaning of probability.

The most familiar example of probability, is the probability of $\frac 12$ for $\text{H}$ and $\text{T}$, each for a single toss of an unbiased coin. This means, that any random toss of a coin (assumed to be unbiased) has equal chances of yielding a $\text{H}$ or a $\text{T}$.

But does it also imply that with a large number of tosses we shall have an equal number of $\text{H}$'s and $\text{T}$'s? Or at least, the general tendency(I don't know the technical term for it) of the experimental result of a large number of tosses is to produce an equal occurrence of $\text{H}$'s and $\text{T}$'s, though for a finite number of tosses there still might be deviations form the expected relative frequency?

Consider for example, a large (say $10^9$) number of tosses. For a unbiased coin, there should be $\frac {10^9}2$ occurrences of $\text{H}$'s and $\text{T}$'s at the end of all the tosses. Say we used a computer to toss all the coins and store the result as a single string of characters $\text{H}$ and $\text{T}$, unknown to us. After the experiment is over, the expected frequency is $\frac 12$ for each outcome. Suppose we start reading the generated string and about halfway, we realize we had much more $\text{H}$'s than $\text{T}$'s.

Then, is the probability of finding the next character to be T greater than that, for finding it to be H? Does the answer change if we know, that after the $10^9$ tosses, the total occurences of both the outcomes were equal, even if we still are unaware of the outcomes of individual experiments (the generated string)?

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See Law of large numbers and Gambler's fallacy. –  Jaycob Coleman Nov 11 '13 at 7:31

2 Answers 2

up vote 1 down vote accepted

The probabilities of outcomes and/or events of a random process are an inherent property of the process and have nothing to do with individual trials or any particular number of trials. Classical probability is the theoretical relative number for an infinite number of trials, or, if you will, the limit of the relative number as the number of trials goes to infinity. Classical probability theory deals with the many interesting situations and questions that arise in the case where all probabilities are known or determined by given conditions.

People have attempted to broaden this concept of probability to include belief etc. but that is not classical probability theory and in fact is really statistics and not probability theory. Statistics deals with situations where a theoretical probability is NOT known or determined. In that case we are limited to estimate or - in special cases that rarely arise - test. (Statistical testing is vastly overused and misunderstood.)

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If you know the underlying probability distribution is 50% heads, 50% tails, it doesn't matter if you flip 100 trillion heads in a row, the next flip will still be 50% heads and 50% tails.

In the real world however, if such an unusual thing were to happen, you would suspect that it wasn't a fair coin. You would say your null hypothesis is that it is a fair coin, then ask, which sequences of flips are "unusual". You might do this 100 trillion flip experiment 50 times. If the first half of one set of 100 trillion flips are heads and the next half weren't, would that count towards being a fair coin or not? You tally all the points for not fair versus fair, then add up the probability that a fair coin would actually give you such an unusual outcome. If a fair coin would only do something so unusual .0001% of the time, chances are it isn't actually a fair coin, and you would say that you reject the null hypothesis (that it is a fair coin).

You might want to look into the binomial distribution with $p=.5$, which would tell you the probability of any number of heads when you flip $n$ coins.

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