# The National Coin Flipping League Championship Series

### The National Coin Flipping League Championship Series

No tech today, but a little basic math.

In baseball, a sport I know little about, apparently the Boston Red Sox have recently come back from a three game deficit to win a best-of-seven series against their traditional rival team, the New York Yankees.

Baseball is a game which attracts statisticians, and many have noted that this is the first time in major league baseball history that a team has won a best of seven series after being down three games to none.

However, it has happened twice in hockey.

I have a modest proposal. Suppose once a year, the National Hockey League and Major League Baseball decide all their various championships without going to all the trouble and expense of playing the game. Rather, they could simply hold a best-of-seven coin-flipping championship. (Call it the Numismatic Hockey League if you'd like.)

Suppose Boston calls heads. The odds of Boston flipping T T T and then coming back to win with H H H H are one in 128.

Therefore, there should be one such occurrence on average every 128 series. There are four such series a year: the American and National League finals, one "world" series (for which only North American teams are eligible, strangely enough), and one Stanley Cup. You'd expect to wait 128 / 4 = 32 years on average between occurrences.

We've been playing pro baseball and hockey, what, about a hundred years in North America?

Three such series, in about a hundred years -- or, roughly one every 32 years. It seems like the math works out rather nicely. Maybe they have been deciding the games via coin flipping and just not telling anyone. Hmm...

Is Boston's victory really that impressive? I mean, the last time I played Risk I rolled three sixes on three dice and England crushed Iceland -- odds of that are 1/216, almost twice as long as Boston coming back from a three tail deficit in the National League Coin Flipping Championship. That's because my blue plastic army guys really worked together as a team and gave 110%!

And yet it didn't make headlines in even the local paper.

In related news, if Houston wins their championship, and it ends up being Texas vs. Massachussets in both baseball AND the presidential election, that's going to be freaky weird. What are the odds of that?

• Drew's analysis is plausible, but completely incorrect. Anyone else want to take a stab at explaining why, or shall I?
• Ah, yes. Not all outcomes are equally likely.
• Right. Chance of HHHH is 1/16, chance of HHHTH, HHTHH, HTHHH, THHHH is 4/32 = 1/8, so you can't count the four-game series and the five-game series as equally likely.
• Eric,

I see noone has commented on you playing Risk.

You really need to acquaint yourself with http://www.boardgamegeek.com ( start with http://www.boardgamegeek.com/top10.htm ) and play some *real* board games.

;-)
• <i>Which raises an interesting point. Winning a series with HTHHTTH is every bit as unlikely as TTTHHHH, but no one ever makes a big deal out of it.</i>

That's because people are perhaps unconciously trying to be statisticians.

This is another aspect of the difference between coin-flipping and baseball. It's already been mentioned that the odds of a Yankees win is any one game is not necessarily 50-50. But more than that, the odds of a Yankees win in any one game is largely unknown. (Certainly not well-defined, at any rate.)

So imagine you're a baseball fan, and you've just watched three games of the Yankees-Red Sox series. Since you're human, you want to be able to predict future events. So you try to guess who's going to win the series, and most likely you try to do so by unconciously estimating the odds that, say, the Yankees will win game four based on what you've seen in games one through three.

The moment you first try to estimate the probability of an event based on past trials, you've entered the realm of statistics. And given the fact that the Yankees have won the first three games, it <i>is</i> statistically valid to infer that the probability of the Yankees winning an individual game is likely higher than 50%. Not much higher...three data points make a poor statistical universe...but likely higher nonetheless. It would be much less valid to make the same inference if the Yankees only win one or two of the first three games.

So if the series ends up the way it did, it will be more surprising to the average fan than if it ends up HTHHTTH. But with good reason: the final outcome was not in agreement with the statistical expectation formed by the first three games. We don't see this effect with coin flips because the probability of heads vs. tails in a coin flip is known from the outset and thus statistics (as opposed to plain old probability) never comes into play.
• No formatting tags, gotcha.
• Basketball also has 7-games series in its playoffs and no team has come back from an 0-3 deficit. IIRC from the TV broadcast, there have been something like 320 cases where a team has gone 0-3 and only 3 have won the next 4 games.
• We were eating lunch, when the restaurant owner, a gambling fan, told us about his latest trip to the casino. He laughed when he told us about a "chump" who, playing roulette, bet on red again after red had already come up three times: "stupid guy, what are the chances of red coming up four times in a row?" Wanting to avoid future food poisoning, we nodded in agreement ...
• Ahh yes, the gamblers fallacy.
• Actually there are 15 7-game series per year in hockey -- every playoff series is a best of 7, and they start with 16 teams.

15 hockey and 3 baseball; I don't know the basketball playoffs (whether there is more than 1 7-game series), so assume 1 for that.

128 / (15 + 3 + 1) = ~6.75.

Since there hasn't been such an upset every 7 years, we can safely assume that they're not flipping coins.
• Basketball has 7 7-game series (the first round, like the first round in baseball is 5 games, but basketball has a 16 team 1st round while baseball has a 8 team 1st round)

so...

64 (HHHTTTT + TTTHHHH) / (15 + 3 + 7) = 2.37

'nuff said
• What would REALLY be interesting is if someone won 7 games emerging from a 6 game Deficit.

• "What would REALLY be interesting is if someone won 7 games emerging from a 6 game Deficit."

The most impressive thing about that would be playing 13 games in a seven game series... :)

Unfortunately by default our brains are hopeless at stistics. The UK national lottery is a 6-balls-from-a-bag type deal, and if the sequence 1-2-3-4-5-6 came up people would be amazed, astounded, and possibly shouting "FIX!", but it's no more unlikely than 7-18-24-29-35-39 and no-one's going to be up in arms if THAT turns up. (Apart from me -- if those are the winning numbers tomorrow I'll be REALLY upset.)

I quite enjoy making people angry with the Monty Hall dilemma (Google for it if you're not familiar with the principle). Some people just refuse to accept the answer matter how many decision trees you draw or practical demonstrations you do as it flies against common sense...
• Ok, flip-heads. Given I spent the past two nights at Busch Stadium watching my Cardinals send the Astros back to their B-hive (by the way, you'd only understand what I'm talking about if you gave a crap about the game) and that in celebrating those two fantastic victories, I tried to do my part in reducing the surplus of Bud Light here in the St. Louis area, by what margin did I defy the odds of making it into work on time this morning?