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Since it is the topic of the day, let's turn it into a statistical question.

Preliminary polls are showing 52 to 48 in favor of staying in the EU in terms of vote.

However bookies are giving a 80%+ chance for UK to stay in EU in their bet odds.

Clearly the bookie odds are linked to the uncertainty in the measurement. If we knew that the result will be 52 to 48 for sure then bookie would assign 100% chance.

Now how do we go from the preliminary poll results to bookie odds with some distributional assumptions, the number of people answered the exit poll (n) and the number of total voters (N), with the simplification that the people answered the poll honestly?

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You want to give confidence interval for the binomial proportion.

Exit poll is like Bernoulli process with the maximum likelihood estimate for the $p$ to be $0.52$, of course.

By central limit theorem this converges to normal distribution and $95%$ confidence interval would be $\hat{p}\pm 1.96*\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$.

So if they asked $1000$ people the real proportion of positive minded people is $(0.48, 0.55)$.

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    $\begingroup$ Actually we are looking for the probability that p > 0.5. $\endgroup$
    – Cagdas Ozgenc
    Jun 23, 2016 at 13:51
  • $\begingroup$ it's low. I don;t think you can give analytical solution. Just checking confidence level completely above 0.5, I suppose. $\endgroup$
    – slakov
    Jun 23, 2016 at 14:07

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