I recently had a harrowing experience on American Airlines, where my traveling companion had a partially collapsed lung and there was a real possibility she would need oxygen on the flight. Luckily American did a great job of getting us home quickly and that was unnecessary. But you have to imagine how it looked to me the very next day when I woke and read this headline:
Woman Dies on American Flight
And I proceed to read that the woman died in the plane with doctors and nurses trying to save her, and where 2 oxygen tanks were empty and the defibrilator didn't work. Needless to say it gave me a chill.
It's the sort of thing that seems like an incredible coincidence, one requiring some sort of explanation other than sheer mathematics. "Perhaps it was a sign from the gods", some would say. "What are the odds?" will ask others. It was the same airline, same problem with breathing, the same day, etc. How can you just say it was a coincidence?
Here's how. Whenever calculating the probability of something occurring, it is crucially important to clearly identify what that something is. Whenever someone says "What are the odds of THAT?", consider what exactly one means by "that". If I roll 10 dice and get 10 1's, and someone says "What are the odds of THAT?", is "that" really just 10 1's? Strictly speaking, yes, and the answer is (1/6)^10, a small number indeed. But in the context of "this is too unlikely to be coincidence requiring some more fanciful explanation", "that" is not just 10 1's. It's also 10 2's, 10 3's, 10 4's etc., perhaps even alternating 1's and 2's, the sequence 1,2,3,4,5,6,1,2,3,4, and a whole host of other results that might prompt the "What are the odds of THAT?" question. For that is what "that" is: not the particular result that you observe, but the set of results that prompts the question.
It is important to remember that the result of any particular roll of the dice is tiny, just like the set of coincidences on plane flights. So saying "the odds of that are very small" doesn't tell us anything we didn't already know before you rolled. It is akin to the difference between someone winning the lottery tomorrow, and the odds of you winning it.
So sure, it seems freaky that it was another woman on the same airlines on the same day with the same problem. But it would also have seemed freaky if they flew from the same city (they didn't), had the same medical condition (not: heart problems vs lung problems), both needed oxygen on the plane, or had both died. There are tons of variables, and yet only a few matched. By only focusing on the ones that match, we greatly increase the probabilities of such seemingly miraculous coincidences, to the point where they are not miraculous at all.