Probability of women TDs

Part of the reform agenda of the last year or so has been the promotion of women in politics and lengthy debates about the introduction of quotas of female TDs or at least quotas of female candidates for election. Those in favour of the status quo usually state that there is nothing wrong with the current system as it elects the best candidates. But does it really?

The Law of Large Numbers describes the result of repeated performance of an experiment. It is the thing that suggests if you flip a coin 100 times about 50 of them should turn up heads. If your sample size is too small, say 10, then the law doesn't hold and results of 2/8 or 3/7 should be expected on a fairly regular basis. Using the binomial distribution you can actually calculate these probabilities. In the 10 flip case a 3/7 split comes up about 12% (120/1024) of the time. This can be figured out on paper using Pascal's Triangle and some basic arithmetic. For larger experiments use of a computer program such as R or using WolframAlpha is recommended.

So back to the Dáil. We start with the assumption that men and women are equally likely to be the best candidate. If the system is fair then each seat should have an equal change of being filled by a female as a male. With 166 seats to be filled, there should be on average 83 of each. However, the sums show that this exact solution will occur only about 6% of the time. So what is the distribution of the rest of the results? Well a 80/86 split is about 5.5%, a 75/91 split is 3%. At a 95% confidence level the result should be between 70 and 96 deputies of each gender. This means that if the result is outside this range there is only a 5% chance of it being due to random factors.

As you reduce the number of female deputies the odds become even longer. 60 or fewer females should only occur 0.02% of the time. 50 or fewer and you are at 1.6x10^-7 or about 1 in six million elections. So where does our measly 23 female TDs come in on the scale? Crunching the numbers reveals a figure of 1.1x10^-22 which is so far beyond random chance that it isn't funny. So the figures indicate a rejection of the hypotheses and we must conclude that either the system is unfair between the sexes or that women are not up to the task of being politicians. Somehow I'm putting my money on the former.