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Abbott’s Cabinet: What’re the chances?

September 26, 2013

Oops! I slipped and fell into politics.

I don’t normally care about politics, but recent events have forced me to. I won’t complain (directly) about all of the terrible decisions by the new government; we’ll just back up some arguments with some basic math. Recently Tony AbbottSee also [1]. revealed his Cabinet, which contains 18 men and 1 woman. Abbott is already well-known for his misogyny, so immediately he was attacked for such an imbalance in the Cabinet. He defends his decision by arguing that we shouldn’t need to ensure a gender balance, and should instead only focus on merit; these are the best 18 people – plus himself – for the job. So what’s the chances (mathematically speaking) that the 18 best people were chosen?

There are obviously a plethora (calender word) of factors that come into play, so let’s calculate the likelihood a few different ways. First let’s naively assume that gender plays no role, and that men and women are equally likely to be chosen for the Cabinet. In this case choosing 17 men and one woman is just as likely as flipping 17 heads and 1 tails. The chances of choosing a man or flipping a head is 50%, or equivalently, a probability of 0.5. The probability of multiple independent things happening is found by multiplying the individual probabilities – the chances of flipping two heads in a row is 0.5 times 0.5 = 0.25 (or 1 in 4). So the chances of choosing 18 men and zero women is 0.5\times0.5\times0.5\times… 18 times, or 0.5^{18}=0.000004=0.0004\% – pretty slim. Now the probability that the first one is a woman and the remaining 17 are men – since it’s equally likely to pick either a man or a woman – will still be 0.5^{18}. But then there’s the possibility of the second member of cabinet being a woman, and the third, etc… So it is 18 times as likely that there is a single woman when compared to zero women; the chances that no more than one woman was chosen is be 0.0004\%+18\times0.0004\%=0.008\%. No bookie in the world would give you odds on that!

But ok, sure. There are other factors to take into account. What if we were to be conservative and say only one in four people who get into politics are women? That’s a pretty conservative estimate right? Well then the chances of an arbitrary politician being male is 0.75, while the probability of them being female is 0.25. So the chances of the entire cabinet being men is now 0.75^{18}=0.006=0.6\% and the chances of one member being female while the rest are male is 18*0.25*0.75^{17}=0.03=3\% – that is, there are 18 ways of choosing one female (0.25) and 17 males (0.75). So even with these conservative estimates for the number of women in politics, the chances that Abbott’s cabinet would turn out this male dominated or worse (without sexism) is less than 4%.

Even if you try to take other things into account, it’s pretty hard to argue the best 18 people for the job were chosen. I’ve even rounded things up prematurely to give an overestimate here.

And with that said, I’ll try to avoid mentioning politics here every again.

  1. Ben Farmer permalink

    Lol, actually by that calculation the cabinet makeup doesn’t seem so surprising to me. It says that “random selection” hypothesis (with your specified weights) is only excluded at p=0.04, which is not very statistically significant.

  2. Ben Farmer permalink

    You would expect nearly 1 in 20 cabinets would have such a makeup if they are just drawing names from a hat :p.

    • Ok sure. But that’s if we assume that 3 out of 4 people who go into politics are men; in reality it’s probably somewhere in between. And it’s not 20 cabinets; it’s the cabinet that Abbott put together.

      Also I could have removed Julie Bishop from it, since she was kind of already guaranteed a spot 😛 then it’s far more damning. I’d say my 4% is generous.

      • Ben Farmer permalink

        Well, it also has to be people who went into politics like 20 years ago, not now. And the 20 cabinets is just to give it a frequentist interpretation; you can be surprised only if a-priori we expected Abbott, or the liberal party generally, to be biased against women (either way it is evidence with a weight of 20:1 for the existence of bias, sure, but I argue that 20:1 odds is not all that conclusive, so your odds have to start out higher than 1:1 for the new evidence to move you to a strong position, i.e. it isn’t enough to swing a rational agent away from even a weak belief that no bias exists)

        But yes your calculation is on the generous side. But also of course we know the selection is not random so it isn’t valid anyway :p. We could perhaps argue something about the distribution of potential political talent being random instead, and then convince ourselves that bias against women in politics generally exists somewhere in their lifetime, whether it is education or social pressure that biases them against it, or bias within the party. To seperate the various possible sources, or to claim one party is worse than another, we’d have to do things like examine the makeup of various past cabinets.

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