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9,223,372,036,854,775,808:1 (not even close)

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9,223,372,036,854,775,808:1 (not even close)

Unread postby furls » Sat Mar 17, 2007 12:39 pm

I originally put this in the Browns thread but thought it should be here. Never underestimate the power of a monkey with a calculator.

9,223,372,036,854,775,808


This is not even close to right. They make a lot of ridiculous assumptions in coming to this number. If I had some time to do the leg work, I could give you a much better probability.

I recreated this number in about 5 seconds by tracing their faulty assumptions. For the math geeks out there, what they did was considered every event as a "Bernoulli Process" with a probability outcome of 50:50 generating a binomial distribution.

Essentially, the calculation becomes (1/2)*(1/2)*(1/2).... 63 times (in order to eliminate 63 teams there must be 63 games in the tournament and 63 outcomes). This generates a % probability, now take the inverse of that to get the "9,223,372,036,854,775,808:1."

The real flaw is that this method assumes that a 16 seed beating a 1 seed has probability of .50 (50:50 chance). In the 20 some years since expanding the tournament this has never happened. That is 80 trials with 0 successes, that implies (statistically) that the probability of a 16 winning the game is 0 and removes 4 powers of 2 from the answer (divide by 16).

Here is another ridiculous aspect of this "model," it assumes (as a consequence of its probability assumptions) that every team has an even probability of winning the tournament. So basically a 16 seed has the same probability of winning the tournament as a 1 (1 in 64 or 1.5625%).
If I offered you any of the 1 seeds this year at 64:1, would you take it? Hell yeah you would. How about if I offered you a 16 seed (or for that matter an 8 seed at 64:1)?

Now you can also essentially remove the 2/15 matchup because the 2 nearly always wins that, and the same with the 3/14 and the 4/13. These upsets happen, but not anywhere near 50% (I would guess 2/15 happens about 3-5% and the others happen about 10%) of the time.

Now all of a sudden instead of 2^63 we actually have ~ 2^47.

These changes bring the probability down from [quote]9,223,372,036,854,775,808 to 1 all the way to 140,737,488,400,000.

Now this 140 trillion is still a big number but there are more simplifications,
That will bring it down futher, outside of the first round, the odds start getting more difficult to calculate as this begins to expand into a Markovian process, but I think you can agree that 1's do not lose to 8/9's 50% of the time, nor do 2's lose to 7/10's 50% of the time and so on.

Statistics like this are exceptionally deceiving and really piss me off because they seem accurate, but they are not. Not even close. With some research, you could reduce the probabilty much further, but it is not worth it for me to do it.... I don't actually get paid to report truthful and accurate facts by a major network (maybe I should).
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Unread postby leadpipe » Sat Mar 17, 2007 4:12 pm

I saw a stat that claimed that if everyone in the world filled out a million brackets each, it would still be 1,000 to 1 shot to hit a perfect bracket. How close do you think those numbers are?
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Unread postby furls » Sat Mar 17, 2007 8:39 pm

Lets do some math here for fun!

probability according to the monkeys with calculators at the San Fran Enquirer:

9.223X10^18

Now estimating the population of the world at 9.223 billion if they all fill out one million brackets, that would be:

(9.223X10^9)(1.0X10^6)=9.223X10^15

Dividing the total probability space (9.223X10^18) by the total number of brackets:

(9.223X10^18)/(9.223X10^15) = 1.0X10^3 = 1000:1 so yes, that is correct.
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Unread postby furls » Sat Mar 17, 2007 8:59 pm

I spent a few minutes on this probability problem I figured out two methods to get a realistic probability of the perfect bracket.

Method 1: Get statistical data for the last 8 years tournaments to establish statistical probabilties and use those in a Markov chain to figure out the probability. I decided against this option because it is time intensive and I think there is a bit more parity now, an effect that would be nullified by the averaging process.

Method 2: The one I chose. I took the largest bracket that I am participating in and took the first round results from the median picker (the guy who is currently 87th out of 174 pickers). He has correctly selected 26 of 32 games giving him a success rate of 81.25%. I assumed (fairly big assumption) that he would regress linearly back to the mean picking percentage at even intervals every round. Basically, I do not think that he will pick through the tournament at 81.25%, that number will go down incrementally from round to round to the monkey throwing darts rate of 50% for the finals. This is because at each stage the teams are going to be "mismatched" somewhat, providing the picker an advantage at picking winners that will slowly disipate as the teams get more evenly matched heading into the final four. Here are how the round by round probabilities came out:

First Round: 81.25%
Second Round: 75%
Sweet Sixteen: 68.75%
Elite Eight: 62.5%
Final Four: 56.25%
Finals: 50%

Now using these round by round probabilties I was able to calculate the overall probability as follows:

Odds of getting 100% of the first round: 0.1301211%
Odds of getting 100% of the second round: 1.0022596%
Odds of getting 100% in the Sweet 16: 4.9909316%
Odds of getting 100% in the Elite 8: 15.2587891%
Odds of getting 100% in the Final Four: 31.640625%
Odds of getting the final correct: 50%

Now, the odds of getting 100% for the whole bracket is the product of all those probabilities which equates to 63,643,669.18:1. That seems alot more reasonable.

Now just for fun, I worked one out for our leader who hit 30 of 32 in the first round for a clip of 93.25%. Here is his table:

Odds of getting 100% of the first round: 12.6788786%
Odds of getting 100% of the second round: 7.4251086%
Odds of getting 100% in the Sweet 16: 11.4266463%
Odds of getting 100% in the Elite 8: 20.7594141%
Odds of getting 100% in the Final Four: 34.515625%
Odds of getting the final correct: 50%

This turns out to a probability of 25,947.57:1. He was rather lucky (I am not sure anyone could really expect that success), but I think that if you were a "expert" you could not do better than our median guy, and I would expect an expert to come in between the two values for the first round (26-30) say 28. At 28, using the same method, the probability was 1,131,460.34:1, and that seems VERY reasonable.

This was not nearly as much work as it seems. It took me 5 minutes to make an excel spreadsheet that did it for me, typing this post took 2X that long.
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Unread postby Jumbo » Sun Mar 18, 2007 11:48 am

Furls,

Some more info that you might find interesting. For the last few years Sean Forman (the guy who made baseball-reference.com) has done a "Monte Carlo simulation" of the tournament run one million times based exclusively on the Sagarin ratings. (So any biases in talent are Sagarin's.)

Here's a link to his page: http://www.60feet6.com/research/talks/NCAA2007/NCAA2007.html

There's an internal link to the summary with probabilities for each team. As an example, OSU recorded a 1.4% chance of losing in the first round, 21.8% the second, 15.3% the third, 30.9% the fourth, 21% the fifth, 4.8% of reaching the finals but losing, and 4.7% chance of winning the tournament.

North Carolina won 45% of the time. 13 teams never won once in a million simulations.

Finally, on his "Conclusion and My Bracket" link, he places the odds of getting all the games correct as one in two trillion. No info on how he gets there.
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Unread postby furls » Sun Mar 18, 2007 12:40 pm

That was pretty interesting thanks for the link. That is definetly another approach to the same problem, I would be very interested in seeing how he got his number and what it means. I would think from the context that he was saying that his bracket, that he chose, had an individual probability of 1:2,000,000,000,000.

There are obvious flaws with the initial run of my simplified model, but it will get better. These flaws are evident by the ridiculous numbers generated by someone who got off to a streaky, hot start.

I am going to use all the pools that I am in to generate a real statistical distribution of picks at each round to get a feel for what % people pick at for each round. I am sure that the numbers probabilities start out higher, I am just unsure how fast they fall off to the monkey throwing darts probability. The problem is a bit more complicated than just taking averages of second round scores, but I am working on it. I am going to update this at the end of every round with my empirical probabilities, and we will see how this all works out.

The Sagarin-Monte Carlo method is fine, but I prefer a method that allows the subjective, non-quantifiable intuitions of pickers determine the overall probability. There are some things that the Sagarin method will not account for, like individual match up problems that are game specific are lost in the Sagarin Rankings (as are current injuries like Brian Butch missing from Wisconsin).
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Unread postby furls » Tue Mar 20, 2007 9:38 pm

I had to make some changes to my approach due to the "pain in the ass factor." It was actually easier, with the help of excel, to work with the mean. I dropped the top and bottom 10% of the group. In my new sample:

Round 1 Probability: 0.746822034

Odds of a perfect first round: 11403.86517:1

Probability of Picking a sweet sixteen team before the tournament started:

0.65095339

Odds of a perfect sweet 16 based on this: 962:1
Coming from a Wolverine, we're the football equivalent of a formerly abused wife of a meth addict who just remarried the safe nice guy. We're just glad we have someone who's aware that it's a rivalry and that tackling on defense is integral. Baby steps.

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Unread postby swerb » Tue Mar 20, 2007 10:21 pm

Odds of me filling out 20 brackets every year and not competing in any of them - 1:8
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