## Super Bowl Coin Flip

Cullen Roche observes,

I was driving all day long yesterday and heard an interesting statistic on the radio. The last 14 coin flips in the Super Bowl have been won by the NFC. The odds of this happening are 16,000:1. Beyond a statistical anomaly. Anyhow, the commentator, a Vegas bookie, was discussing a bet they had going in which they wager on the coin flip every year. He was talking about how silly the bet is because, obviously, the odds are 50% heads or tails. But Vegas is playing it NFC vs AFC and guess what? 75% of the public is betting on the NFC to win the coin flip!

Roche chalks this up to the “recency effect” and implies that it’s a delusion. It strikes me as simple Bayesian logic. Theoretically, the odds of a coin landing on heads are 1 in 2 and that, since either the NFC or AFC will be represented by heads, the odds of either conference winning the coin flip are equal. Forced to bet on which conference will win and given even odds, then, it’s 50-50 that you’ll win.

Yet, in this case, bettors have additional information: The NFC has won the last 14 flips in a row! A 16,000:1 happenstance! That’s probably just a bizarre coincidence. But there’s at least some tiny chance that it’s something else. So, given even odds, why *wouldn’t* you bet on the NFC’s winning again?

I am calling Vegas right now, because if

The last 14 coin flips in the Super Bowl have been won by the NFC. The odds of this happening are 16,000:1.what are the odds of the NFC winning15 coin flipsin a row????The smart bet is on the AFC winning it. Always play the odds.

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Always wondered if the special coins they make for those flips changed the odds a bit.

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At no time in history was there a mathematically fair coinflip.

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Like what?

??? Given even odds, what difference would it make who you bet on?

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@OzarkHillbilly:

50-50.

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@Tano: Shenanigans? An unfair coin? Indeed, the latter would be my guess. That is: the coin’s slightly more likely to land on “heads” because of weighting and maybe the NFC team tends to call heads and the AFC team tends to call tails for some reason.

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There’s also a 16,384:1 chance that the AFC could have won 14 times in a row. Also a 16,384:1 chance that it would have alternated between the two conferences, starting with the AFC. Another 16,384:1 chance that it would alternate but starting with the NFC. Another 16,384:1 chance that it would have been NFC every time a Republican was in the White House and the AFC every time a Democrat was in the White House, etc. Add them all together and pretty soon you have a non-remarkable chance that some pattern will emerge, so the real “problem” is that humans are really good at picking up patterns.

Yeah, the coin undoubtedly has a very tiny bias, but I wonder if you could even *intentionally* make a thin coin-like object that had anything bigger than, say, a 51:49 bias. It’s just too thin for the weight on one side to matter much. And any significant shape cheats (like making it convex on one side to make it tend to roll over, or some aerodynamic device like fins) would be visually obvious.

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If this wasn’t Vegas, and instead this was an office pool where the winners would split the winnings, and 75% picked NFC, then, if you picked AFC and AFC won the coin toss, you would win three times as much.

This is also the reason why, if you would play the lottery and you can select the numbers, you should pick numbers that other people aren’t picking.

Also, 14 in a row is nothing:

Which means, that in the above office pool, if AFC would win this time, you should most likely bet on NFC the next time if you want to maximize your winnings.

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@James Joyner:

Isn’t it a different coin each year?

I’ve always wondered if that would really work (and, of course, if it did, you would have to explain 1) who would do it, why would they do it, how would they get the two teams to choose the right side etc…)

Anyway, I was going to ask if anyone knew whether this has ever actually been tested the I remembered my old friend Mr. Google.

Turns out….You can load a die but you can’t bias a coin – PDF

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@James Joyner:

The coin would have to be so unfair that the former would have to be true too.

Lets say that the side that one team is picking has a 51% chance, then the odds would be 12417:1.

Lets say that the side that one team is picking has a 55% chance, then the odds would be 4314:1.

Lets say that the side that one team is picking has a 72% chance, then the odds would be 100:1.

Lets say that the side that one team is picking has a 95.17% chance, then the odds would be 2:1.

The coin used would have to be extremely unfair.

Add to that the fact that the NFC team hasn’t called it each year. which makes some sort of conspiracy between the head referees and the NFC teams almost impossible.

There isn’t something else here. These things happen.

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@Franklin: Actually, even the penny is weighted more than 51/49. If you’re curious about the coin weightings, try spinning a coin 1000 times on a table. (Though you’ll get bored out of your mind, this removes the mechanical effects of flipping, which are notoriously problematic.) If you really want to do something interesting, though, try flipping the coin 1000 times while always starting with the same side up; the “up” side — no matter what it is — tends to be favored, even if it’s the opposite of what you got from the spinning experiment.

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@ptfe:

Where do you get that from? Sounds completely bogus to me.

If you flip a coin into the air, it revolves at a consistent rate (even as it slows). It does not twirl in some odd rhythm, speeding and slowing, so that one side spends more time facing upward in each revolution. Each side is facing upward equally during each revolution. As a result the side that ends up facing up is determined simply by when the coin contacts the ground, relative to where it is in its last revolution.

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