I got a call from Joe Ax, a reporter at the (Westchester) Journal News because there had recently been two different tied elections in the county. (See here for some links.) He wanted my estimate of the probability of a tied election. Well, there were actually only about 1000 votes in each election, so the probability of a tie wasn’t so low. . . . (For an expected-to-be-close election with n voters, i estimate Pr(tie) roughly as 5/n. This is based on, first, the assumption that there is a 1/2 probability of an even number of votes for the 2 candidates (otherwise you can’t have a tie), and then on the assumption that the outcome is roughly equally-likely to be between 45% and 55% for either candidate. Thus 1/2 x 10/n = 5/n.)
I also mentioned that some people would calculate the probability based on coin flipping, but I don’t like that because it asssumes that everyone’s probability is 1/2 and that voters are independent, neither of which is true (and also the coin-flipping model doesn’t come close to fitting actual election data).
Coin flips and babies
An hour or so later Joe called me back and said that he’d mentioned this to some people, and someone told him that he’d heard that actually heads are slightly more common than tails. What did I think of this? I replied that heads and tails are equally likely when a coin is flipped (although not necessarily when spun), but maybe his colleague was remembering the fact that births are more likely to be boys than girls.
P.S. Here’s the Journal News article (featuring my probability calculations).
Actually, the comment about coin flips being biased is correct. See the article at: http://news-service.stanford.edu/news/2004/june9/…
Briefly, a mechanical coin flipper was created and it was found that there is a slight bias towards heads. Though in normal situations with a person flipping a coin that bias likely dissappears (bouncing on the ground, tossed at different heights with changes in rotation).
It also depends on the coin! There was a funny debate on whether the Euro coins are biased: Euro coin accused of unfair flipping.
Scott,
No. The cited article says that there is a 51% chance the coin will land "the same way it's started", not a 51% chance of heads. Deb Nolan and I discuss this in our article (cited in the above post): depending on the method of flipping, the probability of landing on the same side as started can be varied. But, if the coin is flipped high enough and with enough uncertainty that the initial state is "forgotten," then the probability of "heads" will necessarily be 50%.
This is assuming the coin is flipped and caught in the air–not spun, not allowed to bounce.
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Aleks,
We mention the Euro coin in our article. The short story is: no bias is flipped, possible bias if spun.
Great problem for my undergrad math stats students!
I use a hierarchical model with the number of votes, X, having a binomial distribution, and the probability of "success", p, having a beta distribution with a mean of 0.5 and a standard deviation of 0.025 (so +/- 5% is 2s.d.). A little integration and a little calculation gives an interesting result: for N=1000, Pr(X=500) = 0.0134653. Since 1000 is even, this should be about twice your estimate — and it's darn close.
Aleks,
Ahh, yes, that is what I get for reading the article I linked to too quickly and not your paper!
I have also found the coin flip to be a great way of explaining the multiple comparisons/testing problem (brain imaging research, hundreds of thousands of simultaneous tests…much easier to start with fair coin flips)
cheers,
Scott