The Probability of Everything

Numbers always seem to carry the weight of authority. The thinking, at least subliminally, goes like this: if a teacher awards grades on a 100-point scale, those tiny distinctions must really mean something. But if ten publishers could deem the manuscript for the first Harry Potter book unworthy of publication, how could poor Mrs. Finnegan (not her real name) distinguish so finely between essays as to award one a 92 and another a 93? If we accept that the quality of an essay is somehow definable, we must still recognize that a grade is not a description of an essay’s degree of quality but rather a measurement of it, and one of the most important ways randomness affects us is through its influence on measurement. In the case of the essay the measurement apparatus was the teacher, and a teacher’s assessment, like any measurement, is susceptible to random variance and error.

Voting is also a kind of measurement. In that case we are measuring not simply how many people support each candidate on election day but how many care enough to take the trouble to vote. There are many sources of random error in this measurement. Some legitimate voters might find that their name is not on the rolls of registered voters. Others mistakenly vote for a candidate other than the one intended. And of course there are errors in counting the votes. Some ballots are improperly accepted or rejected; others are simply lost. In most elections the sum of all these factors doesn’t add up to enough to affect the outcome. But in close elections it can, and then we usually go through one or more recounts, as if our second or third counting of the votes will be less affected by random errors than our first.

In the 2004 governor’s race in the state of Washington, for example, the Democratic candidate was eventually declared the winner although the original tally had the Republican winning by 261 votes out of about 3 million.1 Since the original vote count was so close, state law required a recount. In that count the Republican won again, but by only 42 votes. It is not known whether anyone thought it was a bad sign that the 219-vote difference between the first and second vote counts was several times larger than the new margin of victory, but the upshot was a third vote count, this one entirely “by hand.” The 42-vote victory amounted to an edge of just 1 vote out of each 70,000 cast, so the hand-counting effort could be compared to asking 42 people to count from 1 to 70,000 and then hoping they averaged less than 1 mistake each. Not surprisingly, the result changed again. This time it favored the Democrat by 10 votes. That number was later changed to 129 when 700 newly discovered “lost votes” were included.

Neither the vote-counting process nor the voting process is perfect. If, for instance, owing to post office mistakes, 1 in 100 prospective voters didn’t get the mailer with the location of the polling place and 1 in 100 of those people did not vote because of it, in the Washington election that would have amounted to 300 voters who would have voted but didn’t because of government error. Elections, like all measurements, are imprecise, and so are the recounts, so when elections come out extremely close, perhaps we ought to accept them as is, or flip a coin, rather than conducting recount after recount.

This is an extract from Leonard Mlodinow's The Drunkard's Walk published by Pantheon