The National Coin Flipping League Championship Serieshttp://blogs.msdn.com/b/ericlippert/archive/2004/10/21/the-national-coin-flipping-league-championship-series.aspxNo 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 Yankeesen-USTelligent Evolution Platform Developer Build (Build: 5.6.50428.7875)re: The National Coin Flipping League Championship Serieshttp://blogs.msdn.com/b/ericlippert/archive/2004/10/21/the-national-coin-flipping-league-championship-series.aspx#571115Sat, 08 Apr 2006 00:20:24 GMT91d46819-8472-40ad-a661-2c78acb4018c:571115MSDNArchiveThere 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.
<br>
<br>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.<div style="clear:both;"></div><img src="http://blogs.msdn.com/aggbug.aspx?PostID=571115" width="1" height="1">re: The National Coin Flipping League Championship Serieshttp://blogs.msdn.com/b/ericlippert/archive/2004/10/21/the-national-coin-flipping-league-championship-series.aspx#250345Mon, 01 Nov 2004 01:50:00 GMT91d46819-8472-40ad-a661-2c78acb4018c:250345Norman DiamondI 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.
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<br>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.
<br>
<br>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.
<br>
<br>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.<div style="clear:both;"></div><img src="http://blogs.msdn.com/aggbug.aspx?PostID=250345" width="1" height="1">Hoo-eee!http://blogs.msdn.com/b/ericlippert/archive/2004/10/21/the-national-coin-flipping-league-championship-series.aspx#249397Fri, 29 Oct 2004 07:20:00 GMT91d46819-8472-40ad-a661-2c78acb4018c:249397rubikzube*<div style="clear:both;"></div><img src="http://blogs.msdn.com/aggbug.aspx?PostID=249397" width="1" height="1">re: The National Coin Flipping League Championship Serieshttp://blogs.msdn.com/b/ericlippert/archive/2004/10/21/the-national-coin-flipping-league-championship-series.aspx#248853Thu, 28 Oct 2004 01:24:00 GMT91d46819-8472-40ad-a661-2c78acb4018c:248853Norman DiamondI don't think Mr. Lippert explained the envelope problem.
<br>
<br>> "You either gain X (=Y) by switching, or
<br>> lose 1/2 X (=Y) by switching."
<br>> X and X/2 can't _both_ be Y in the same
<br>> sentence!
<br>
<br>True but so what? The "quoted" sentence has this meaning:
<br>"Either X = Y or X/2 = Y. If you open the envelope then you know the value of X but you don't know the value of Y until later.[*] You either gain X (and later find out that X = Y) by switching, or lose X/2 (and later find out that X/2 = Y) by switching."
<br>The paradox is not in that explanation.
<br>
<br>The actual explanation was determined by some participants in rec.puzzles a few years ago. I almost remember enough math to understand it. There is no such thing as a uniform distribution over the set of integers.
<br>
<br>I can add a bit to the explanation. Pretend that the assumed probability distribution (uniform over the set of integers) were possible. Your expected gain is infinity divided by infinity, or infinity minus infinity, however you want to express it.
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<br>[* Yes, part of the apparent paradox is that you don't have to open the envelopes in order to decide that you want to switch back and forth an infinite number of times. Nonetheless it is true that IF you open your envelope THEN you will know X while still not knowing Y. This still explains why the possibility of either X matching Y or X/2 matching Y is not a paradox.]<div style="clear:both;"></div><img src="http://blogs.msdn.com/aggbug.aspx?PostID=248853" width="1" height="1">re: The National Coin Flipping League Championship Serieshttp://blogs.msdn.com/b/ericlippert/archive/2004/10/21/the-national-coin-flipping-league-championship-series.aspx#248356Wed, 27 Oct 2004 05:16:00 GMT91d46819-8472-40ad-a661-2c78acb4018c:248356KC LemsonI helped an elderly gentleman win thousands of dollars at a casino last year... I rolled double 6's at one point. He looked me in the eye, said "I trust you, I know you'll do it again" and put $50 on it... I rolled it again. He put another $50 on it, and I rolled it a third time. He collected his wad of dough and then left, thanking me - with words.<div style="clear:both;"></div><img src="http://blogs.msdn.com/aggbug.aspx?PostID=248356" width="1" height="1">re: The National Coin Flipping League Championship Serieshttp://blogs.msdn.com/b/ericlippert/archive/2004/10/21/the-national-coin-flipping-league-championship-series.aspx#247785Tue, 26 Oct 2004 07:54:00 GMT91d46819-8472-40ad-a661-2c78acb4018c:247785Dan ShappirEric, one of the better answers that I got. Good for you!<div style="clear:both;"></div><img src="http://blogs.msdn.com/aggbug.aspx?PostID=247785" width="1" height="1">re: The National Coin Flipping League Championship Serieshttp://blogs.msdn.com/b/ericlippert/archive/2004/10/21/the-national-coin-flipping-league-championship-series.aspx#247653Tue, 26 Oct 2004 00:09:00 GMT91d46819-8472-40ad-a661-2c78acb4018c:247653Eric LippertThe Envelope Paradox gets into knots because of the probabilities. So let's state it without probabilities.
<br>
<br>Statement 1: Let the amount in your envelope be X. You either gain X by switching, or lose 1/2 X by switching. Since it's obviously better to gain X than lose 1/2 X, you should switch.
<br>
<br>Statement 2: Let the amount in the smaller envelope be Y. You either have Y or 2Y. If you switch, you might gain Y or you might lose Y. There's no benefit to switching.
<br>
<br>Clearly at least one of these statements is wrong because they contradict each other. But now the misdirection is obvious:
<br>
<br>"You either gain X (=Y) by switching, or lose 1/2 X (=Y) by switching."
<br>
<br>X and X/2 can't _both_ be Y in the same sentence!
<br><div style="clear:both;"></div><img src="http://blogs.msdn.com/aggbug.aspx?PostID=247653" width="1" height="1">re: The National Coin Flipping League Championship Serieshttp://blogs.msdn.com/b/ericlippert/archive/2004/10/21/the-national-coin-flipping-league-championship-series.aspx#247426Mon, 25 Oct 2004 21:15:00 GMT91d46819-8472-40ad-a661-2c78acb4018c:247426Eric LippertIs this what you C&O people do for fun?
<br><div style="clear:both;"></div><img src="http://blogs.msdn.com/aggbug.aspx?PostID=247426" width="1" height="1">re: The National Coin Flipping League Championship Serieshttp://blogs.msdn.com/b/ericlippert/archive/2004/10/21/the-national-coin-flipping-league-championship-series.aspx#247424Mon, 25 Oct 2004 21:12:00 GMT91d46819-8472-40ad-a661-2c78acb4018c:247424ZacBut what if instead of picking heads or tails, Boston and New York each have to pick a sequence of three outcomes (e.g. HHT), and New York has to pick first, and whoever's sequence appears first wins ...
<br>
<br>:)
<br><div style="clear:both;"></div><img src="http://blogs.msdn.com/aggbug.aspx?PostID=247424" width="1" height="1">re: The National Coin Flipping League Championship Serieshttp://blogs.msdn.com/b/ericlippert/archive/2004/10/21/the-national-coin-flipping-league-championship-series.aspx#247244Mon, 25 Oct 2004 15:59:00 GMT91d46819-8472-40ad-a661-2c78acb4018c:247244Eric LippertThis is a fairly well-known paradox. Google "Newcomb's Paradox" for another.
<br>
<br>The resolution of this paradox lies in the fact that you're using "X" to mean two different things in one sentence. In one half of the sentence, X represents the larger amount, and in the second half it represents the smaller amount, so of course they're different.
<br>
<br>Stop using "X" and start using dollar amounts and the paradox goes away.<div style="clear:both;"></div><img src="http://blogs.msdn.com/aggbug.aspx?PostID=247244" width="1" height="1">re: The National Coin Flipping League Championship Serieshttp://blogs.msdn.com/b/ericlippert/archive/2004/10/21/the-national-coin-flipping-league-championship-series.aspx#247100Mon, 25 Oct 2004 10:57:00 GMT91d46819-8472-40ad-a661-2c78acb4018c:247100Dan ShappirIn the spirit of this thread, here's an interesting logical/statistical conundrum: Suppose I offered you to choose one of two identical sealed envelopes, telling you that both contain an unspecified amount of money, but that one contains double the amount of the other. Obviously the choice is random.
<br>
<br>Now suppose that before you open the envelope I give you the chance to exchange it. If we assume your envelope contains X $, then the other envelope contains either half of X or double X. So, you’ve got a 50% chance of loosing 1/2 X, and a 50% chance of gaining X. So it would seem that contrary to reason, it makes sense to exchange the unopened envelopes.
<br>
<br>And say I offered you the chance again, and again, and again. It would seem that we would stand there forever, exchanging the unopened envelopes. What’s wrong here?<div style="clear:both;"></div><img src="http://blogs.msdn.com/aggbug.aspx?PostID=247100" width="1" height="1">re: The National Coin Flipping League Championship Serieshttp://blogs.msdn.com/b/ericlippert/archive/2004/10/21/the-national-coin-flipping-league-championship-series.aspx#246956Mon, 25 Oct 2004 00:03:00 GMT91d46819-8472-40ad-a661-2c78acb4018c:246956AnonEric,
<br>
<br><quote>
<br>You'll note that I did not mention WHEN I last played Risk. It was probably eight or ten years ago.
<br>
<br>How is it then that I so clearly remember my blue armies in England crushing Iceland? Because my Risk strategy is:
<br>
<br>* always play blue, and
<br>* get into a huge, endless fight over Europe, sapping my strength until eventually the guy who's held Russia/Asia for the last ten turns cashes in 120 armies worth of cards (card limits are for wimps!) and sweeps across Europe in a bloody rampage.
<br>
<br>So at some point England ALWAYS crushes Iceland with triple sixes.
<br>
<br>It's not an _effective_ winning strategy, but it's fun. Particularly if you can make a last stand on Iceland with a hundred armies or so.
<br></quote>
<br>
<br>ROFL...<div style="clear:both;"></div><img src="http://blogs.msdn.com/aggbug.aspx?PostID=246956" width="1" height="1">re: The National Coin Flipping League Championship Serieshttp://blogs.msdn.com/b/ericlippert/archive/2004/10/21/the-national-coin-flipping-league-championship-series.aspx#246653Sat, 23 Oct 2004 13:42:00 GMT91d46819-8472-40ad-a661-2c78acb4018c:246653Eric LippertFlawed at its VERY CORE, eh?
<br>
<br>Your first complaint is the same one I addressed forty comments ago. Don't like the odds? Pick different odds. Weight the coin differently. The chances of getting a TTTWWWW series change, but not by much.
<br>
<br>Your second point is also easily dispensed with. Suppose for instance a team has a .5 chance of winning the first game, and then a .75 chance of winning the next game if they won the previous game. In such a system, the chances of a TTTHHHH series go UP, not DOWN! Pick any conditional probability model you want, it's only going to make such events more common.
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<br>No, the real flaw is in the validation. The model is insufficiently complex to explain on probabalistic grounds why there have been so few TTTWWWW series given that there are more seven game series than I stated.
<br>
<br>We could come up with a new model in which all these concerns were met. Divide all pairings of teams into "thoroughly outclassed" and "about even".
<br>
<br>In the vast majority of "thoroughly outclassed" matchups, one team has such a higher probability of winning that four-game series are practically inevitable.
<br>
<br>But in a few "about even" matchups, we should expect to see a proper coin-flipping distribution. About 1/8 of them should be sweeps, about 1/64 of them should be three losses followed by four wins, etc.
<br>
<br>As argued in the comments above, clearly Boston and the Yankees were in the "about even" category. <div style="clear:both;"></div><img src="http://blogs.msdn.com/aggbug.aspx?PostID=246653" width="1" height="1">re: The National Coin Flipping League Championship Serieshttp://blogs.msdn.com/b/ericlippert/archive/2004/10/21/the-national-coin-flipping-league-championship-series.aspx#246629Sat, 23 Oct 2004 10:00:00 GMT91d46819-8472-40ad-a661-2c78acb4018c:246629OrenI sense a great disturbance in the force... as if billions of Statistics professors cried out at once, and then fell silent. Your attempts to analyze a sports event using this statistical model is flawed at its core:
<br>1. In statistics, we assume that a coin toss has exactly a 50% chance of each outcome. This is not true in a sports game.
<br>2. In statistics, a series of coin tosses is "memory-less". The coin doesn't remember what happened the last time it was tossed, so it has the same 50% chance of each outcome on every toss. Humans do have memory.
<br>
<br>Those are the two mathematical errors in your model. There are also errors in your analysis, such as the fact that there are far more 7-game series than you had mentioned, but that's not a mistake in the model; only in its validation.
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<br>In summary: never trust the words of a man who writes in a purple font...<div style="clear:both;"></div><img src="http://blogs.msdn.com/aggbug.aspx?PostID=246629" width="1" height="1">Vegas and Mathhttp://blogs.msdn.com/b/ericlippert/archive/2004/10/21/the-national-coin-flipping-league-championship-series.aspx#246495Fri, 22 Oct 2004 23:45:00 GMT91d46819-8472-40ad-a661-2c78acb4018c:246495Bloga de Nathan<div style="clear:both;"></div><img src="http://blogs.msdn.com/aggbug.aspx?PostID=246495" width="1" height="1">re: The National Coin Flipping League Championship Serieshttp://blogs.msdn.com/b/ericlippert/archive/2004/10/21/the-national-coin-flipping-league-championship-series.aspx#246442Fri, 22 Oct 2004 18:14:00 GMT91d46819-8472-40ad-a661-2c78acb4018c:246442Eric LippertOne certainly hopes so.
<br>
<br>Between May and August of this year I was averaging 200-300 Google hits a day.
<br>
<br>It was over _600_ yesterday, and some of those have got to be gems.<div style="clear:both;"></div><img src="http://blogs.msdn.com/aggbug.aspx?PostID=246442" width="1" height="1">re: The National Coin Flipping League Championship Serieshttp://blogs.msdn.com/b/ericlippert/archive/2004/10/21/the-national-coin-flipping-league-championship-series.aspx#246431Fri, 22 Oct 2004 18:04:00 GMT91d46819-8472-40ad-a661-2c78acb4018c:246431Brian ButlerThis topic should make some interesting contributions to "Riddle Me This, Google: Part Three".<div style="clear:both;"></div><img src="http://blogs.msdn.com/aggbug.aspx?PostID=246431" width="1" height="1">re: The National Coin Flipping League Championship Serieshttp://blogs.msdn.com/b/ericlippert/archive/2004/10/21/the-national-coin-flipping-league-championship-series.aspx#246406Fri, 22 Oct 2004 17:35:00 GMT91d46819-8472-40ad-a661-2c78acb4018c:246406Eric Lippert> I see noone has commented on you playing Risk.
<br>
<br>You'll note that I did not mention WHEN I last played Risk. It was probably eight or ten years ago.
<br>
<br>How is it then that I so clearly remember my blue armies in England crushing Iceland? Because my Risk strategy is:
<br>
<br>* always play blue, and
<br>* get into a huge, endless fight over Europe, sapping my strength until eventually the guy who's held Russia/Asia for the last ten turns cashes in 120 armies worth of cards (card limits are for wimps!) and sweeps across Europe in a bloody rampage.
<br>
<br>So at some point England ALWAYS crushes Iceland with triple sixes.
<br>
<br>It's not an _effective_ winning strategy, but it's fun. Particularly if you can make a last stand on Iceland with a hundred armies or so.
<br><div style="clear:both;"></div><img src="http://blogs.msdn.com/aggbug.aspx?PostID=246406" width="1" height="1">re: The National Coin Flipping League Championship Serieshttp://blogs.msdn.com/b/ericlippert/archive/2004/10/21/the-national-coin-flipping-league-championship-series.aspx#246324Fri, 22 Oct 2004 15:30:00 GMT91d46819-8472-40ad-a661-2c78acb4018c:246324FryguyOk, 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?
<br>
<br>Head.....Hurts.....<div style="clear:both;"></div><img src="http://blogs.msdn.com/aggbug.aspx?PostID=246324" width="1" height="1">re: The National Coin Flipping League Championship Serieshttp://blogs.msdn.com/b/ericlippert/archive/2004/10/21/the-national-coin-flipping-league-championship-series.aspx#246305Fri, 22 Oct 2004 14:47:00 GMT91d46819-8472-40ad-a661-2c78acb4018c:246305Mat Hall"What would REALLY be interesting is if someone won 7 games emerging from a 6 game Deficit."
<br>
<br>The most impressive thing about that would be playing 13 games in a seven game series... :)
<br>
<br>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.)
<br>
<br>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...<div style="clear:both;"></div><img src="http://blogs.msdn.com/aggbug.aspx?PostID=246305" width="1" height="1">re: The National Coin Flipping League Championship Serieshttp://blogs.msdn.com/b/ericlippert/archive/2004/10/21/the-national-coin-flipping-league-championship-series.aspx#246295Fri, 22 Oct 2004 14:11:00 GMT91d46819-8472-40ad-a661-2c78acb4018c:246295JorgeWhat would REALLY be interesting is if someone won 7 games emerging from a 6 game Deficit.
<br>
<br>Think about THAT.<div style="clear:both;"></div><img src="http://blogs.msdn.com/aggbug.aspx?PostID=246295" width="1" height="1">re: The National Coin Flipping League Championship Serieshttp://blogs.msdn.com/b/ericlippert/archive/2004/10/21/the-national-coin-flipping-league-championship-series.aspx#246277Fri, 22 Oct 2004 13:15:00 GMT91d46819-8472-40ad-a661-2c78acb4018c:246277simsBasketball 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)
<br>
<br>so...
<br>
<br>64 (HHHTTTT + TTTHHHH) / (15 + 3 + 7) = 2.37
<br>
<br>'nuff said<div style="clear:both;"></div><img src="http://blogs.msdn.com/aggbug.aspx?PostID=246277" width="1" height="1">re: The National Coin Flipping League Championship Serieshttp://blogs.msdn.com/b/ericlippert/archive/2004/10/21/the-national-coin-flipping-league-championship-series.aspx#246267Fri, 22 Oct 2004 12:47:00 GMT91d46819-8472-40ad-a661-2c78acb4018c:246267EricActually there are 15 7-game series per year in hockey -- every playoff series is a best of 7, and they start with 16 teams.
<br>
<br>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.
<br>
<br>128 / (15 + 3 + 1) = ~6.75.
<br>
<br>Since there hasn't been such an upset every 7 years, we can safely assume that they're not flipping coins.<div style="clear:both;"></div><img src="http://blogs.msdn.com/aggbug.aspx?PostID=246267" width="1" height="1">re: The National Coin Flipping League Championship Serieshttp://blogs.msdn.com/b/ericlippert/archive/2004/10/21/the-national-coin-flipping-league-championship-series.aspx#246257Fri, 22 Oct 2004 12:16:00 GMT91d46819-8472-40ad-a661-2c78acb4018c:246257TadAhh yes, the gamblers fallacy.<div style="clear:both;"></div><img src="http://blogs.msdn.com/aggbug.aspx?PostID=246257" width="1" height="1">re: The National Coin Flipping League Championship Serieshttp://blogs.msdn.com/b/ericlippert/archive/2004/10/21/the-national-coin-flipping-league-championship-series.aspx#246210Fri, 22 Oct 2004 08:55:00 GMT91d46819-8472-40ad-a661-2c78acb4018c:246210Dan ShappirWe 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 ...<div style="clear:both;"></div><img src="http://blogs.msdn.com/aggbug.aspx?PostID=246210" width="1" height="1">