Here’s an amazing statistic, coming from the insightful TV commentary leading up to kick off in yesterday’s Super Bowl.

It came after the large no-cash-value metal disc they loosely called a “coin” was flipped into the air and landed on the side they’d decided that today they would refer to as “heads”. I think that side had a helmet on it, so it was close enough.

“The past 13 years the NFC team has won the toss. The odds of one team winning 13 straight are about eighty-one hundred to one”.

Obviously, this kind of revelation is the reason commentators don’t STFU during the pre-match shenanigans and wait for the actual game to start.

In fact, the number he was looking for is closer to eighty-two hundred. Two to the power of thirteen is 8,192, which would give odds (assuming a fair fake coin) of 8,191-1.

The number’s right, but it’s not quite the right context. Indeed, across a series of Super Bowls, the odds of the team from one particular chosen conference to win 13 coin flips is as above, but the probability of one team winning 13 straight is lower.

If we don’t care who wins the first toss (after all, someone has to win it) then whoever does only needs to guess another 12 correctly to make a streak of 13. The odds of that happening are 4095-1.

Still a fairly unlikely occurrence, but it’s half the first number – and I’m sure they’d still have mentioned it if was the AFC team who’d got there.

It would be just as noteworthy – in fact, probably more so – if there was a streak of 13 straight heads or 13 straight tails. The odds of any one of the four streaks I’ve mentioned so far happening brings the price down to a heavily discounted 2047-1.

I began wondering, with all the stats that are thrown about during a typical NFL game, whether a game ever passes that something doesn’t show up that looks vaguely remarkable.

This coin toss streak actually only considers consecutive Super Bowls. It’s much easier to find wonderful patterns if you consider consecutive playoff games, or Monday evening games, or games played in domes, or in the rain, on the West Coast where at least one team is wearing blue. And they do look for this kind of meaningless correlation. All the time.

But in this case it didn’t need any such fudging. I’ve tried to discount it as much as I can, but it’s still a pretty terrific streak of coin flips coming down in the same arbitrary (yet definitely consistent) direction.

It’s one to remember the next time you hear someone complaining about a “sick” (see my air quotes there?) losing streak when playing online poker. Maybe they lost five or six 50/50 races in a row.

They’ll probably try to convince you this never happens in real life. Guess what… it actually does.

For what it’s worth, if you’d bet a dollar on the NFC winning the coin toss in 1998 and parlayed the winnings onto the same bet for the past 13 years at the typical bookmakers odds of 10-11, you’d be in for a whopping payout of $4474.51.

Not too shabby – but that equates to juice of more than 45% when you compare it to the true odds payout of $8192!

It’s human nature to spot patterns in random events. AIUI that’s because we evolved to connect causes and effects as a survival mechanism – there’s a big rumbling sound getting louder and coming from that mountain -> run away fast &c &c.

There’s something similar in the treatment of anxiety attacks, ISTR they call it something like negative reinforcement theory – “today I got out of the left side of the bed and cracked my head on the door, and I didn’t have an attack” becomes the ritualistic “if I don’t bash my head against the door after getting out of bed on the left side I’ll have a panic attack”.

Of course consciously it’s a ridiculous notion but this stuff is all hardwired deep into the subconscious. And the worst thing is it becomes self-fulfilling so in a way it’s true.

I won’t get started on religion or alternative medicines….

Then again, the question should really be, what are the chances of there being a sequence of 13 in 43 tosses (the number of superbowls). I’m pretty sure that that’s significantly less than 2000:1

Given odds of a particular event = 1/2, odds of a run of 13 events is (1/2)^12, or 1/4096; however, given 31 sequences of 13 flips, that means you have odds of (4095/4096)^31, or just under 99.25%, of the sequence _not_ occurring in the 43 flips.

So I make that just over .75%, or about 132:1.

Does that sound right?

I get a similar answer an easier way 🙂 The streak could have started (or ended) on 32 possible Super Bowls (this year’s was XLIV = 44), 4096/32 is 128 so that’s 127-1. Powers of 2 are nice. Would be 131-1 with 31 games, LOL rounding error-aments.