Tuesday, April 03, 2007

Dialogue Involving A Question of Statistics

Fred Vaughan headshot by Fred Vaughan

"I've got a question, Ray. Everyone knows your opinions about miracles and that it's all physics, so what I want to know is how your good fortune can be reconciled with statistical probabilities. What you have experienced seems impossible to me, Ray!"

"It's the wrong use of the telescope again, Tim."

Tim listened and laughed. "How can that be? Where's the telescope?"

"Well, you're asking about the probability of an event after it has already occurred, aren't you? So probability and statistics don't apply. Statistics has to be directed the other way. Let's say I flip a fair coin one hundred times and get one hundred heads in a row. How would one square that with statistics? Isn't that the essence of your question?"

"Yes! That is the question, Ray! How is it you could flip one hundred heads in a row?" Tim affirmed.

"Well, what if I had flipped a head and a tail alternating until I had fifty heads and fifty tails? Would that bother you as much?" Ray asked.

"No, of course not! That's fifty-fifty, right on the law of averages!" Tim said.

"Well, you're using your telescope incorrectly then, because both cases — a hundred heads in a row, and a sequence of head-then-tail fifty times in a row — have exactly the same likelihood. The only reason you think the one case more likely is because it's similar to a kazillion other cases that are also fifty-fifty. But whatever combination of heads and tails that you get after a hundred flips of that coin will be exactly the same likelihood as the hundred heads, Tim. You flip a coin a hundred times and whatever sequence of heads and tails that you get will have been exactly that unlikely. But a lot of them are disguised."

" Disguised? You've got to be kidding me, Ray!" Tim was not convinced.

"Nope. I'm not." Ray seemed to be done with that discussion.

Tim came back with, "Wait, Ray! That makes no sense! This kind of thing just doesn't happen!"

Ray seemed somewhat tired as he replied, "Your key phrase there was 'kind of thing', Tim. Classes of situations like flipping fifty heads in one hundred flips of a fair coin are the 'kind of thing' that are phenomenally more likely than flipping all heads or all tails. But what you're missing here is that each one of those situations like, head-tail-tail-tail-head-head-tail-head… etc., is no more likely than flipping all heads. There are just more situations that comprise the class involving fifty heads. There are kazillions of them like I said. Does ten-to-the-twenty-ninth have any meaning for you, Tim? Remember! Whenever something actually happens, it is a single situation not a class of them. Everything is unlikely, Tim. Everything! When you flip your coin a hundred times, whatever you come up with will have defied odds of ten-to-the-thirty-first-to-one! But don't doubt for a second whether what happened actually happened, or if it defied the laws of physics, just because of that or you'll be legally insane. Something happens! It has to."

Tim looked as baffled as an ostrich blinking at a bright sun.

Ray knew he had been a pompous asshole. He had indeed been phenomenally lucky. He had to admit that much. Wasn't 'fair' coin defined as one for which one hundred heads in a row does not happen? What about each subsequent flip of that coin along the way? Any one of those coming up tails would have terminated the phenomena of Ray Bonn. Ray Bonn was not some metaphysical being standing back behind a protective glass watching the coin flipping; he was the coin flipping. He was the outcome of all the contingent coin tosses; anything else was an instance of that most major of logical fallacies, looking down the wrong end of telescopes.

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