When you have over 750,000 miles on just one of the big carriers, you are always aware that with each flight the odds get better of a problem occurring. Air travel is very safe, but the actuarial tables don’t lie. Fly often enough long enough and something will certainly happen and after awhile you wonder “ is this the one that does me in”?
This is what's known as the "Gambler's Fallacy" — the notion that if you've flipped tails five times in a row, the next flip is more likely to go heads.
In fact, each coin flip is independent: the coin doesn't know what has happened in the past.
Now, other than making fun of a common fallacy — certainly a worthy effort in itself — there's another reason to post this. As I was thinking about it, a way to point out the problem with this reasoning occurred to me that I had never thought of before. (Original? I doubt it. But new to me.)
Think about the whole population on the plane. In fact, assume that Frank, with his 750,000 miles, is sharing the plane with some five year old who has never been on a plane before in his life. If this reasoning were correct, the odds of the plane crashing have to be much higher for Frank ... but much lower for the five year old.
Since the plane can't both crash for Frank and not crash for the five year old, there has to be something wrong with the original reasoning.