When is improbability a proof of tampering? So, I just watched a video of some guy being accused of cheating in a game (he tried to place a world record) and the argument boiled down to "this sequence of events is so astronomically improbable that we declare that you cheated".
Which is OK, fair point. However... it got me thinking. There are many ways that something can be really improbable. If I shuffle a full deck of 52 cards, the odds of THAT particular sequence are on the order of 1 in 8*1067. But just because I got it doesn't mean I cheated, and I don't think anyone would accuse me of it either.
Or, let's put it in a more abstract way. Let's take a dice roll. A standard, 6-sided dice. If you roll it 100 times and get a value of "6" in all of them, you'd probably be suspected of cheating because the odds of that happening are $\cfrac{1}{6^{100}}$. On the other hand, if you got the values 33535516343413352544 35425515536356234615 62256526346236661225 31422624135261426131 54445152412651245623, the odds of getting that sequence are also $\cfrac{1}{6^{100}}$. So why isn't this suspicious?
I feel like I'm missing something trivial.
 A: Short answer: It comes down to patterns people perceive and the number of configurations that can be associated with those patterns. In the space of all possible configurations, configurations with a pattern comprise a small part of that space, and thus such a pattern is unlikely to occur randomly.

Long answer: You're right that any specific sequence of 100 die rolls is just as likely to be drawn as a sequence where a $6$ is drawn for each of those rolls. However, the fact that we could draw two sequences of the former type and-at a glance-be unlikely to distinguish between them suggests that our mind is unable to intuit any pattern or order that defines either specific sequence. In fact, our mind perceives both sequences as random as is evident in your case by the fact that you likely just wrote an unplanned sequence of digits, and if you were to write the question again you would just write another (likely different) unplanned sequence of digits. But if you were to repeatedly write this question, you would easily quote the 100 die rolls of $6$ each time.
This is to say that your mind perceives an order in the $100$ die rolls of a $6$ that is does not perceive in the specific and equally probable $100$ die rolls for a particular sequence. It perceives this specific sequence as random in much the same way that clothes thrown about a room have specific positions but appear randomly placed to an onlooker.
And so when it intuitively evaluates the probability of such a sequence, it is not asking "What is the probability that I get a $3$, then another $3$, then a $5$, then another $3$,...?" but instead asking "What is the probability that I get a random collection of numbers?" Conversely, with the $100$ rolls of a $6$, your mind does see a specific pattern thus asks "What is the probability that I get $100$ $6$s in a row?"
If we take $\Omega_A$ to be the number of $100$ die-roll sequences with any random number of $1$s, $2$s, $3$s, $4$s, $5$s, and $6$s, and $\Omega_B$ to be the number of 100 die-roll sequences with only $6$s, it is clear that $\Omega_A \gg \Omega_B$. Thus, with $\Omega_{\text{tot}}$ being the number of all such die-roll sequences, it is also clear that the probability of a random sequence $p_A = \Omega_A/ \Omega_{\text{tot}}$ is much greater than that for the "only $6$s" sequence $p_B = \Omega_B/ \Omega_{\text{tot}}$.

Dream drama: I don't know anything about the "Dream drama" mentioned in an above comment, but if someone was accused of cheating due to there apparently being a low probability of something, it would be because this something is associated with a pattern that the accusers can perceive and such a pattern is unlikely in the space of possible configurations of what they are observing.
