## What are the odds of Sen. Franken?

In a comment on Janet's post, Ironman points out a statistical study of the King County recount and its effect on the gubernatorial race in Washington, where we saw a recount reverse a previously announced result. Ironman provides a calculator to test the results of this change from the recount which, it is argued, is just a resampling of the ballots. (Indeed, there's a test audit being done of the Coleman-Franken vote already, using 202 of the more than 4000 precincts in the state, which is simply part of the state's quality control.) The error rate on machine read ballots is reported to be about 0.2%, or two in a thousand. That's why we recount with a margin under 0.5%.

The calculator works for a binomial distribution. What complicates this is the presence of Dean Barkley as a holder of 15% of the ballots that are being sampled. So it's drawing from a bag with three different colored balls, not two different colors. (I'm sure there were third-party candidates in the Gregoire Rossi '04 race, but I doubt it would have made much difference.) If you assume no change in the Barkley votes, however, you still get a less than .005% chance that a recount would flip this election to Franken, even with that .2% error rate. If the erroroneous ballots are random and the population of ballots are divided roughly fifty-fifty, the probability is like flipping a coin say 5800 times and getting 236 more heads than tails. As you flip the coin more, the probability converges on 0.5 quite quickly. But again, that makes some assumptions about the distribution of the errors regarding the inclusion of Barkley, and to be blunt I'm not a good enough statistician to think that part through. Ironman, back to work!

And to the rest of you, spreadsheets please if you have running totals for any county. You can potentially look for the problems with recounting that with a good formula and those sheets.

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