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Independence and Randomness

While no reader actually submitted this question, it was presented to me while I was chatting with someone online. The question goes something like this: Suppose you have a ten-sided die (0-9). You want to create four-digit numbers with it, so you will roll it four times and mark down those results down, and that will be one four-digit number. The question is whether numbers with repeated digits will be less frequent.

When I first heard this, I wasn't exactly sure what was meant. What does "less frequent" mean? It turns out that he meant less frequent than you would expect. How often do we expect a number with the same four digits to occur, anyway? Let's assume that we would allow 0000 (zero) to be a "four digit repeat." That is, if our first roll is a zero, we would not have to re-roll because we are allowing any digit to be any number. In that case obviously there are only ten possible four-digit repeats. And there are obviously 10,000 possible outcomes (0-9999), so one in a thousand of our random numbers on average would be a four-digit repeat.

The idea that four-digit repeats would be less likely is somewhat absurd, really, once you consider a wonderful test for a very basic idea in statistics, that of independence. Of course, independence has a definition, and it's not very hard to grasp, but the test I suggest is to ask yourself, "Does the ___ know what came before?" In this case, does the die know what rolls came before? No, so each roll is independent and the probabilities we calculated above are accurate. (If we are dealing out cards without replacement, does the deck "know" what came before?)

And think about this: a four-digit repeat has no meaning to dice. It is only in our own minds that we can look at a number and see significance like that. Would one expect one's birthday to turn up less often than expected because it's such a special number?

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