What if you flipped a coin 20 times, and got one of these results? (H stands for heads, and T stands for tails.)
Now, there is only one chance in a million of getting sequence 1. You would look at it and say immediately that something was wrong -- the coin was somehow loaded, or something.
There is, however, only one chance in a million of getting sequence 2, as well. There is only one chance in a million of getting any particular sequence. So there is nothing suspicious about getting a particular sequence unless that sequence is distinctive.
Which leads us to sequence 3. It is distinctive. Is it distinctive enough?
This question has real-world analogues. Probability tells us what the chances are of something happening before it does. We generally, however, want to know after it happened how significant that occurence is.
I choose three volatile stocks. In July, I mail out a prediction that the Smith Company will rise in August to 4,000 potential investors; I mail out a prediction that it will drop in August to 4,000 other potential investors. One of those predictions is correct. To half of those to whom I mailed the correct prediction, I mail the prediction that the Jones Company will rise in September, to the other half that it will fall in September. One of those predictions is correct. To half of those to whom I mailed the correct prediction, I mail the prediction that the Brown Company will rise in October, to the other half that it will fall in October. That gives me 1,000 prospects for my stock-prediction newsletter who know that I've been correct on my last three picks.
If the police get a DNA sample at the scene of the crime, then they can check the DNA of a suspect, or even a dozen suspects. The likelihood of a false positive is less than one chance in a million. There is, however, now a bank of millions of DNA samples on file. The probability of a false positive among them approaches one.
In politics, we have a history of Democratic administrations seeing a higher growth than Republican administrations do. Is that a coincidence, or is that a pattern? This is one of the debates of the current period.