## Probability brain teaser of the day

Over at Andrew Gelman's, a reader emails:

I read the abstract for your paper What is the probability your vote will make a difference? [...] I'd note that the abstract prima facie contains an error. Your sentence in the abstract, "On average, a voter in America had a 1 in 60 million chance of being decisive in the presidential election." can not be correct. If we assume that this sentence is correct that means that given the actual turnout of 132,618,580 people the sum total probability of voters being decisive is larger than one. This of course [sic] is impossible. The total amount of decisiveness must be at most one.

What is going on here? Can the probabilities sum to more than one? Take a minute to think it through, and then read the rest of the entry to find out the answer...

The above argument is at first appealing but is not actually correct. Actually the total probability can exceed 1.

The reason the total probability can exceed 1 is that it is possible for many voters to be decisive at the same time.The original blog post is here.