The Ridiculously Low Probability Of Mitt Romney's 8-Vote Victory In Iowa

Mitt Romney maths

Photo: Tim Scott via Flickr

After the most attended Republican Iowa conference ever, and 122,000 ballots cast, the margin separating the top two candidates was seemingly negligible: 8 votes.Mitt Romney ended up pulling off a win against Rick Santorum. Business Insider consulted with Clifford Hurvich, a statistics professor at New York University’s Stern School of Business and a fellow of the American Statistical Association.

Santorum was gaining momentum rapidly in the run up to he caucuses, so Hurvich considered two scenarios. The first followed data compiled just days before the election by the Des Moines Register, one of the more accurate and respected surveyors going into the Iowa caucuses. The second scenario equally matched the two men.

First, using data from the Register. The paper reported that 24% of voters favoured Romney to Santorum’s 15% — Ron Paul was between the two with 22%.

At the time 51% of respondents said they were already decided, while 41% said they could be persuaded and the remaining 8% accounted for those who were undecided or who had no first choice.

Based on the Register, Santorum should have received about 11,000 votes fewer than Romney.

For example, let’s isolate Santorum and Romney, and use only their votes: This gives a population n of 60,022 and a p, or the probability a candidate will vote for Romney, of 0.615.

You calculate that as follows:. 24/(.24+.15)=.615. Those figures are taken from the percentage of votes each man was expected to receive by the paper.

To find the probability that the difference between the two is eight or less, we need to calculate Prob{30,007 <= X <= 30,015}, or the sum of each probability: [p(30,007) + p(30,008) … + p(30,014) + p(30,015)].

Using statistical software to calculate the area of those probabilities under a curve with p of 0.615, produces a p-value of near 0, or a 0% likelihood of occurring.

“The p-value is 10^{-708} which represents the probability that we would obtain results as close or closer than those actually obtained yesterday if in fact Romney was as far ahead of Santorum as indicated by the earlier poll,” Hurvich says. “Since this p-value is so small, it leads us to reject this null hypothesis, and start to believe instead that the two candidates are now running neck and neck.”

To put that in perspective, that’s a decimal with 707 zeros followed by a one.

But, even considering Santorum’s surge over the final days, eight votes or fewer separating the two is still highly unlikely.

If X is the number of votes for Romney in a random sample, then X has a binomial distribution with p=.5 and n=60,022.

Summing those probabilities we get 0.029, or 2.9% probability that two equally matched candidates would be separated by at most eight votes.

Think of it as a coin toss, with the probability that you’ll receive a heads or a tails on any given toss with equal likelihood. The chance that the difference is that close in reality is surprisingly low because the coin can easily ride to one side or the other.

But, it still can happen. And as of press time, no candidate had requested a recount. There are less likely things that happen everyday.

“The probability is higher than the probability of drawing three of a kind in a 5-card poker hand, which is .023, or 2.3%,” Hurvich says.

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