Significance Intrudes on Contestants Row
We worked out the likelihood that there would be only one clean sweep, with all six contestants getting on stage coming from the same seat in Contestants Row, out of six thousand episodes of The Price Is Right. That turned out to be not terribly likely: it had about a one and a half percent chance of being the case. For a sense of scale, that’s around the same probability that the moment you finish reading this sentence will be exactly 26 seconds past the minute. It’s pretty safe to bet that it wasn’t.
However, it isn’t particularly outlandish to suppose that it was. I’d certainly hope at least some reader found that it was. Events which aren’t particularly likely do happen, all the time. Consider the likelihood of this single-clean-sweep or the 26-seconds-past-the-minute thing happening to the likelihood of any given hand of poker: any specific hand is phenomenally less likely, but something has to happen once you start dealing. So do we have any grounds for saying the particular outcome of one clean sweep in 6,000 shows is improbable? Or for saying that it’s reasonable?
Perhaps the past 6,000 episodes were short in clean-sweep episodes. There’s no reason that would keep clean sweeps from turning up very often in the next 6,000 episodes. For that matter, there’s no reason there couldn’t be six clean sweeps in the following one thousand episodes. That would be surprisingly many, but it would put us on track for seven clean sweeps in 7,000 episodes, as expected a value as possible.
Imagine if you flipped four coins and they all came up tails: there’s nothing suspicious about that. Happens all the time. Well, it happens one-sixteenth of the time, but that’s close enough to all for our purposes. Flipping eight coins and having them all come up heads … that’s much less likely, but it doesn’t seem quite impossible yet. It’s less likely than this one-clean-sweep thing, and I admit I’d probably get bored and move on to some other activity before getting eight tails on eight coins, but it’s still imaginable. Flipping forty coins and seeing them all come up tails? That’ll never happen and we know it.
Except, how do we know that? There’s nothing physically impossible in having forty fair coins all come up tails. It’s not a though the first 39 heads will gang up on the last coin and force it to come up heads. It’s just that it’s so unlikely to happen that we’d be suspicious of seeing the result. Somewhere between the four coins all coming up heads and the forty coins all coming up heads we find an outcome we just can’t believe was due to chance.
This is the introduction to statistical significance.