Bayes’ theorem lets us answer this if we know just two numbers (well, actually four, but two are just one minus the others).
First, what did you think was the probability of success in Afghanistan before the mission began? This is the prior probability, which we’ll call Ps. The probability of failure, Pf, is one minus this.
Second, what is the probability that we’d see the number of deaths we have, if the mission were succeeding? Call this Pd|s. One minus this gives us Pd|f.
Let’s pin some numbers on this. Let’s call Ps 0.8. I guess this is the sort of number the government had in mind when the mission began. It can’t be much lower, because no-one would send troops to war without a high probability of success. But it can’t be much higher, because that would be over-confident, given that the British and Russian empires both failed to control Afghanistan.
Our second question is trickier. It could be that the mission was always going to be bloody, and in fact we’ve had fewer casualties than expected. On this view, Pd|s is high, and the deaths are no evidence of mission failure - in fact, the opposite. On the other hand, there was a vague hope that troops would “leave without firing a shot.” From this perspective, Pd|s is low, and the deaths are evidence of mission failure.
Let’s split the difference, and call Pd|s 0.5 so Pd|f is 0.5.
Now let’s take Bayes’ theorem to derive a number for the probability of the mission succeeding, in light of the casualities we‘ve suffered. This is Ps|d, the posterior probability:
Ps|d = (Pd|s x Ps)/[(Pd|s x Ps) x (Pd|f x Pf)]
Plug the numbers into this, and we get:
Ps|d = (0.5x0.8)/[(0.5x0.8) + (0.5 x 0.2)] = 0.8
In other words, the posterior probability is no different from the prior. The reason for this is simple. I’ve constructed the example so that the deaths are no evidence either way. So there’s no reason for the prior probability to change.
But what if the deaths were more consistent with failure? If Pd|s were, say, 0.2 and Pd|f 0.8, then our posterior probability would be 0.5.
Now, I write this not to take a view on Afghanistan. I know as much about Afghanistan as Abu Hamza does about playing the banjo. I do so rather to show how Bayes’ Theorem can help us think about the issue. I’d stress two uses of it.
First, it forces us to ask of any piece of information: what is this evidence of? Are deaths evidence of mission failure or not? It’s in this context that we need specific military expertise.
Second, we can reverse engineer the theorem. Rather than plug numbers into it and accept the answer, we can use it to ask: what numbers give us a high probability of success? How plausible are they? Again, military expertise can illuminate here.
Note that I am not saying that Bayes’ theorem is sufficient. It can’t define “success”. Nor can it tell us whether the success is worth the costs. It is, though, surely an important part of how to analyse the issue. And yet - in public at least - politicians do not think this way. They did not tell us Ps or Pd|s (or even ranges thereof) before the mission began. Nor do they tell us now. We can all think of reasons for this omission. But are they really satisfactory?
