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I'm looking for a name of topic(s), theories which describe the situation provided below:

Let's say we have a 6-sided fair dice. And after 6000 throws we have the following results:

  • Digit 1: 1145 times
  • Digit 2: 1089 times
  • Digit 3: 614 times
  • Digit 4: 1132 times
  • Digit 5: 1022 times
  • Digit 6: 998 times

According to the theory of probability the chance of having any digit in the next throw is 1/6 which is fair enough, but when I look at the results of the previous throws I assume that I should expect more throws of digit #3 soon, cause according to the law of large numbers the fair dice results should balance over time. So I'm looking for a way to calculate / measure how the fact may affect the probability of getting digit 3 in the next throw.

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    $\begingroup$ I searched earlier but didn't locate a suitable duplicate. Also see here and here for more on the gambler's fallacy. $\endgroup$
    – Glen_b
    May 28 '19 at 4:55
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According to the theory of probability the chance of having any digit in the next throw is 1/6 which is fair enough, but when I look at the results of the previous throws I assume that I should expect more throws of digit #3 soon, cause according to the law of large numbers the fair dice results should balance over time.

This is an entirely mistaken notion, but it does have a name - it is a form of the gambler's fallacy. The law of large numbers does not lead to any "balancing" over time, where probabilities shift to compensate for previous deviations from expected results.

There's nothing that acts to "compensate" for an excess in the count over subsequent counts (how could there be? the die has no memory of its history). Let's say I roll a fair die 6000 times but I discover I have 45 more "1"'s than you expect with a fair die. Will I expect fewer than 1000 "1" results in the next 6000 tosses? No, on a fair die, my expected number of "1" outcomes is unchanged, it's still 1000.

Note that this means that at the end of 12000 rolls, given I had an excess of 45 on the first 6000 rolls, I still expect to have an excess of 45 rolls. This doesn't contradict the law of large numbers.

Indeed the expected distance between the count in any of the outcomes and the expected count grows without bound as $n\to\infty$ -- quite the opposite of what many people think. It just grows more slowly than $n$ does (as $n^\frac12$ for large $n$). As such, after you divide by $n$ to get sample proportions, those count-deviations are scaled down to something that shrinks as $n$ increases. So the proportions do indeed converge to their expected values, as $n\to\infty$, because the absolute deviations in counts from expected don't grow at the same rate as the $n$ in the denominator.

What actually happens with earlier discrepancies in probability? They are simply "washed out" in the long run (as $n\to\infty$) by the effect of dividing by $n$. There will also be later discrepancies in probability, but they, too are washed out in the long run, just not in any sense by a tendency to compensate for them.

As such, there's no modification to the $\frac16$ probability for a fair die -- that remains as it was, no matter what the size of the deviation from expected in your sample was. [Instead, usually a considerable deviation from expected in the sample would cause you to revise your assumption of fairness, rather than the independence of rolls.]

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  • $\begingroup$ Thank you, that's very comprehensive! $\endgroup$
    – Mike
    May 28 '19 at 3:10

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