Flipping Coins and Gender Quotas

Brendan O'Dowd
Applying physics to politics

Brendan J. O’Dowd is a postdoctoral researcher in the School of Physics in Trinity College, and is studying electricity networks. 

Next year’s general election will see the introduction of gender quotas, wherein Irish political parties will be obliged to ensure that at least 30% of their candidates are female. The plan is to curb the glaring gender imbalance in Dáil Éireann, where women occupy only 27 of the 166 seats, or a little over 16%. Whether the scheme is a good idea is the subject of on-going debate; some see it as reverse-sexism and think that the women who are elected might be undermined if parties are “forced” to put them forward, others see it as a necessary step towards levelling the playing field.

As I physicist, I don’t want to wade in to the rights and wrongs of this approach, but the numbers are interesting to examine. Just how uneven is the playing field? How unusual is our current situation? What would a reasonable ratio look like and how far away are we from that?

If we assume that men and women are equally capable of running the country, then the odds of any seat being won by a man or a woman is 50/50, or the same as tossing a coin. So the ratio of men and women in Dáil Éireann could be decided by tossing 166 coins, or the same coin 166 times. Intuitively, we can guess that the most likely outcome is 83 male, 83 female. We could probably also guess that the odds of the Dáil being completely full of just men or just women would be very low, since that would be similar to getting 166 heads or tails in a row. The exact odds of these events, and every other possible arrangement, is defined by a mathematical function called a “binomial distribution”. In this specific case, the binomial distribution tells us that the odds of there being X women among 166 people is given by:


The second term here is the number of ways of choosing X objects from a set of 166, and is a fairly simple, well-defined function that is commonly used in the area of combinatorics and probabilities. The shape we get for our probability distribution is a peak centred at 83 and getting very close to zero at the edges.


Now, it’s not particularly useful to simply look at the odds of there being exactly 27 female TDs. This is because the odds of a particular outcome depend very strongly on the size of the sample set, and even the most probable outcome can seem to be very unlikely in large datasets. It’s more illuminating to add up all the odds within a particular range. Here we add up the probability of there being 27 women, 26 women, 25 women, all the way down to the probability of no women at all. In the graph, this means that we are adding all the blue data points together. This technique is called a ‘binomial test’, and is often used by statisticians to determine if an outcome can be reasonably expected, or if some external forces must be presumed to be at work.

When we add up all the blue data points here, we find that the probability of there being 27 female TDs or less in a fair vote is very small indeed: 0.00000000000000001% (1.14 x 10-19). Taking the inverse of this, we can see that the chances of this happening is about 1 in 9 quintillion (a quintillion is a billion billion) (8.8 x 10-18). If we had an election every second of every day non-stop, we still wouldn’t expect this to happen over the lifetime of the universe. If election for men and women were equally likely, we can tell from the distribution that we should expect the number of women elected to be between 71 and 95 about 95% of the time. Having less than 67 women should be a very rare event (<1%).

The proposed rules don’t relate to sitting TDs, however, just the proportion of candidates standing for election (and even then it’s just political parties, not independents). So let’s look at the 2011 General Election, in which there were 566 candidates. Using a similar approach as above, we find that the probability of only 30% of these being female (under fair circumstances) is even smaller than 27 women in the Dáil, a thousand times less in fact: 0.00000000000000000004%, or 4.5 x 10-22.

So even if we do achieve a 30% proportion of female candidates in 2016, the situation is still completely out of kilter with what you’d expect in a fair society. Maybe just acknowledging this is a good place to start when thinking about a remedy.