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Superman's product of the century (so far):
I promised further analysis on the manual recount focused on King County.
I think the best way to talk about it at this point is to look at probabilities. To do that it will be instructive to go back to the machine recount briefly (the first recount). These are the final certified results for King County for the machine recount:
|Gregoire Post||Gregoire Pre||New Gregoire||Rossi Post||Rossi Pre||New Rossi|
Let's evaluate this in a probability context. For our purpose we will look at this as a binomial distribution. What this means is that we are only going to look at the pool of Gregoire and Rossi voters - so all of our math will ignore the third party candidate and any other kinds of ballots (like over votes and under votes). This simplifies the mathematics but the results won't be any different. This is the rather daunting looking binomial distribution formula:
Where the (n over x) value is n! (n factorial) over x!(n-x)! - or n!/x!(n-x)!
Stay with me now. Fortunately, we don't have to slog through the whole calculation by hand. Programs like Microsoft Excel have a built in function called BINOMDIST which will calculate the probability solution of a binomial problem for us.
So let's set up the problem: For the machine recount, assume that we have a large bag of marbles, they are either red (Rossi votes) or blue (Gregoire votes), there are a total of 856,963 (505,836 blue + 351,127 red) marbles in the bag (it's a BIG bag of marbles) - these numbers are inclusive of the 'new' ballots discovered (336 apparently), or 'enhanced' during the process.
These marbles are uniformly distributed in the bag - like people, they are all mixed up together and there is no formulaic method to tell where one of the 336 new ballots came from or what precincts or demographics characterize those ballots that required 'enhancing'.
Let's set up the BINOMDIST function:
BIOMDIST(Number of successes in trials, number of trials, success probability per trial, cumulative - TRUE, FALSE supplied value to get either a cumulative probability (up to the number of successes - TRUE) or the actual probability of the number of successes occurring - FALSE.
In our case the numbers are:
a) Number of successes in trials: 593, the number of blue marbles (Gregoire votes) taken from the bag in total.
b) Number of trials: 941, the total number of blue marbles (Gregoire votes) and red marbles (Rossi votes) taken from the bag.
c) Success probability per trial, The probability of reaching into this bag and pulling out a blue marble is 59.03% (505,836/(505,836 + 351,127)) - which corresponds to the overall Gregoire percentage win in King County.
d) Cumulative value in this case is FALSE - we want the actual probability for the case at hand - this is the worst case scenario.
The result? The probability of reaching in this bag of marbles and pulling out 941 of them and having 593 of them be blue is: 0.12%. That's odds against of 853 to 1. Not impossible but very improbable - if we performed this test 853 times we would expect to get this result once.
Now, let's evaluate the manual recount. Once again, it is a good assumption to say that we don't have any kind of demographic or precinct formula to describe the 566 'new' ballots that were counted - they were a random sample of ballots that happened to have signature problems. Nor is there any particular way to determine how additional over votes were converted to voter intent by the Canvassing Board. The King Country manual recount results:
|Gregoire Post||Gregoire Pre||New Gregoire||Rossi Post||Rossi Pre||New Rossi|
So let's solve the binomial distribution for the manual recount in King County:
a) Number of successes in trials: 358, the number of blue marbles (Gregoire votes) taken from the bag in total.
b) Number of trials: 537, the total number of blue marbles (Gregoire votes) and red marbles (Rossi votes) taken from the bag.
c) Success probability per trial, The probability of reaching into this bag and pulling out a blue marble is 59.03% (506,194/(506,194 + 351,306)) - which corresponds to the overall Gregoire percentage win in King County - these numbers are so massive that the recount additions can do little to change the overall percentages - which makes this mathematical solution especially compelling.
d) Cumulative value again in this case is FALSE - we want the actual probability - again the worst case.
What's the manual recount result? The probability of reaching in this bag of marbles and pulling out 537 of them and having 358 of them be blue is: 0.0048%. That's odds against of 20,695 to 1. This is much more improbable than the machine recount -if we performed this test 20,695 times we would expect to get this result once.
There's much more to this story though. The machine recount and the manual recount can be viewed as individual probabilistic events - we can easily calculate the odds of these two events happening back to back (i.e. in sequence). This is given by the probability of the first event multiplied by the probability of the second event.
We can frame the question this way: What are the odds of taking two random samples from a very large bag of marbles back to back with the total ratio of marbles in the bag being 59.03% blue and 40.97% red - in the first sample pulling 941 marbles and having 593 of them be blue, and in the second sample pulling out 527 marbles and having 358 of them be blue? The answer is 0.12% x 0.0048% = 0.00000567%.
The odds of these two improbable events happening back to back is 17,650,517 to 1.
Perhaps it is no coincidence that the Washington state lottery site has 'Election Results' listed at the top center of its site. The odds of winning the Lotto game in Washington (picking 6 of 49 numbers) are 13,983,816 to 1 (you get two chances per one dollar ticket).
What King County and the Democratic party would like us to believe is that Christine Gregoire is the recipient of good fortune that is 20% more improbable than a Lotto win. That one county produced two sets of recount data, reported that data after all the other counties had reported, and just randomly found enough 'votes' to overturn an election.
Cynics might offer that people actually do win the Lottery. They might suggest that this is just the way things happened, that simply because it is this way - it just happened. Against all odds, it just happened. They forget that when the Lottery is won it is the result of millions and millions of chances taken by millions of people so that the odds are matched by the number of sets of numbers selected.
The only certain way to win the Lotto is to buy 6,991,908 tickets and pick all 13,983,816 sets of numbers. The only certain way to overturn the Washington gubernatorial election through recounts was to do so with explicit human intervention.
Cynics might also offer that statistics and probabilities are just numbers games and don't mean too much. They don't realize that billions of dollars are risked and invested in the insurance business, in the commodities markets, in environmental regulations, and the reliability of the products that are bought by billions of consumers - all based on calculations like these. They don't realize that calculations like these are used as a basis for government policy in such diverse things as Middle East peace, war games, and interest rates.
Now, it is also the case - and really the more realistic case - that the number of blue marbles we pull out of the bag could be any of a number of values and calling for them to be specifically the number that King County came up with belies that they could be within a group of numbers. This is certainly true and can be thought of this way: For the machine recount, 83% of the time when we draw the marbles out of the bag we will pull out between 535 and 576 marbles for each trial. Almost 8% of the time we will pull out 534 or less marbles out and about 7% of the time we will pull out less than 592 marbles for each trial. We would only pull 593 or more marbles out a total of .67% of the time in a trial. For the manual count those numbers are that over 99% of the time we will pull out between 287 and 347 blue marbles per trial. We would only pull 358 marbles or more out 0.0164% of the time. Giving every possible benefit of the odds being at least the number that King County produced for each recount in this way still produces an overall odds against of 898,272 to 1. Not quite the odds of the lottery but is it a bet you would ever take for any reason? And I'm not talking about a $1 lottery ticket. I mean something valuable to you something that would hurt if you lost - like what value would you put on democracy? Would you bet living in a nation, in a state where government was by the will of the people? Do you know anyone on the planet that would take that bet with these odds?
The partisan Democrats and individuals at King County who are behind this are willing to play the odds that no one will pursue and gather the evidence that will expose their actions. Do you like being played by
King Ukraine County?
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I couldn't resist - I took your instructions and posted an election recount odds calculator over at my blog. Thanks for the idea!
EMCEE: Cool. Note that making the 'exact' number is a little specious on my part - note the next to last paragraph for the 'realistic' odds (sum of probabilities for the 'exact' number OR greater).
Posted by: Ironman at Dec 29, 2004 10:50:33 AM
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