# 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?

• I would like to expand a bit on that. As already mentioned, the explanation that was determined by participants in rec.puzzles was: There is no such thing as a uniform distribution over the set of integers. An explanation of the explanation can be explained a bit better than I did.

Based on the fallacious assumption that there is a uniform distribution over the set of integers, one can compute that one's expected gain from switching is infinite. Similarly one can compute that one's expected gain from switching back to the original envelope is also infinite. Also the expected amount of money in both envelopes is infinite, and the expected gain is infinity minus infinity.

Actually the game can be simplified (though I haven't seen anyone do it). There is just one envelope. A probably malicious hacker persuades you that the amount of money was chosen at random from a uniform distribution over the set of all integers. You compute that the expected amount of money in the envelope is infinite. Watch out, the Sultan of Brunei and someone else might be tailing you.

This is not much different from fallacies involving division by zero. 1*0 == 2*0 therefore 1 == 2. Well, it was true that 1*0 == 2*0.
• There is a better way to answer this than by trying to remember your statistics classes:  write a simulation.  Even highly educated people get all bent out of shape about the Monty Hall problem and its ilk, and are only convinced by writing programs to demonstrate the effects of random chance.

Given enough seasons, you should actually be able to come up with a better statistical model than any particular guesses (which are always colored by memory, themselves) would grant.  And given that model, you can find out where the cross-over point with the RISK scenario lies.  That is, if you can demonstrate that the model is reasonable.
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