Accidents Will Happen?

Accidents Will Happen?

by Oct 20, 2015Prosecutor's fallacy0 comments

On October 3, 2015 the US bombed a hospital run by the NGO Medecins Sans Frontieres (MSF), killing 22 and injuring dozens. This is an especially bad incident for the US, possibly constituting a war crime depending on the exact circumstances. From the beginning US officials have virtually ruled out the possibility that the hospital strike was intentional. For example, the top US commander in Afghanistan said:

“A hospital was mistakenly struck. We would never intentionally target a protected medical facility.”

Senator John McCain, a State Department spokesman and a White House spokesman all expressed similar certainty that the bombing was unintentional.

They may all be right. However, I would argue that the certainty of these four men is inappropriate and may be explained in part by a common conceptual trap known as the prosecutor’s fallacy.

To grasp this point we first need to lay a bit of ground work on the concept of conditional probability. Role a fair die and we know it will come up 5 with a probability of 1/6. Suppose now that I role a die, hide the result from you but tell you it is an odd number. My disclosure eliminates 2, 4 and 6 from the running, leaving behind 5 as one of just three equally likely possibilities. Thus, if you know the die has landed odd, the probability of rolling a 5 is equal to 1/3. This is also called probability conditional on odd, because the die by assumption satisfies the condition that it has come up odd.

Conditioning works by eliminating possibilities and reallocating their probabilities among the remaining ones. When conditioning eliminates all of the most likely states of the world then the scene left behind by the conditioning operation can present itself to us as if it were a Martian landscape. For example, at the moment the probability that Lindsey Graham will become US president is pretty close to zero. But the probability that Lindsey Graham will become president conditional on receiving the Republican nomination might not be extremely far from ½, or one-in-two. The same might be said of Lincoln Chafee on the Democrat side.

Who would win a Graham-Chafee matchup in 2016? This question would flummox (temporarily) even the most experienced political junky since no one has thought seriously about this question. To put ourselves in a world in which these candidates win the primary votes is to change the electorate substantially from what it seems to be currently, and predicting their preferences in that case seems an extremely difficult task.

You are, of course, evading the Graham-Chafee question if you say either that Graham would win because Chafee will never get the nomination or that Chafee would win because Graham will never get the nomination. To answer the question that was asked we must consider Graham versus Chafee conditional on both winning their party’s nomination.

The prosecutor’s fallacy is, at its core, a mistake of conditioning probabilities on the wrong premise. A disturbing example is the Sally Clark case in the United Kingdom. Ms. Clark suffered the extreme misfortune of having both her small boys die shortly after their births and, still worse, of being convicted of killing them. The British prosecution argued that either the two children died spontaneously or Ms. Clark killed them, and then noted that as it is extremely rare for two children to die spontaneously, Ms. Clark must have killed them. But this one-sided presentation of the evidence misses the point that it is also extremely rare for a mother to kill her children. Thus, the defense could have argued, with roughly equal strength to the prosecution’s case, that Ms. Clark was innocent because a mother almost never kills her two children.

If we combine both sides of the argument we arrive at the banal observation that double deaths of a mother’s children are exceedingly rare, whatever the causes. This truism provides little leverage to assess guilt or innocence on those rare occasions when a mother’s two children actually do die. Presuming guilt under this scenario would be roughly equivalent to presuming that Chafee would beat Graham if they are both nominated because Graham is extremely unlikely to be nominated in the first place. In other words, the prosecutor’s fallacy substitutes an answer to a wrong question for the answer to the right question.

To produce coherent analysis we need to condition our probabilities on the right thing. The miscarriage of justice inflicted on Sally Clark stemmed from calculating probabilities from the (wrong) vantage point of when her children were alive rather than from the (right) vantage point of after they were already dead. Beforehand any reasonable person would have thought, as the prosecution argued, that it was extraordinarily unlikely that the two children would soon die of their own accord. But this fact is almost exclusively because two young children rarely die so suddenly of any cause. Nevertheless, after the unthinkable happened it suddenly became plausible that they had died on their own.

We now return to the American air strike on the MSF hospital in Kunduz. This excellent article by Jon Schwartz attempts to list all the major cases of US air attacks on civilian facilities going back to the First Gulf War. He finds eleven such attacks including ones on Red Cross facilities (not hospitals), on media facilities and on a Chinese Embassy. This history suggests that prior to the attack on MSF it was reasonable to judge a US attack on a hospital as extremely unlikely—but not impossible. By implication, both intentional and unintentional attacks on hospitals were also extremely unlikely before the MSF attack actually happened.

Now that the unthinkable has happened we commit the prosecutor’s fallacy (in reverse to the Sally Clark case) if we jump to the conclusion that it must be a unintentional because we simply cannot believe that the US would stoop so low as to intentionally target a hospital. Doing so ignores the countervailing fact that prior to the attack the probability of unintentionally hitting a hospital was also extraordinarily low (as was the probability that Sally Clark would murder her two young children). The right question without other information about the attack might be: if there’s an attack on a medical facility, what’s the probability it is a intentional? This conditional probability may be low, but it is worth trying to estimate. We have now landed on a Martian landscape where things that looked nearly impossible a few short days ago have become live possibilities.

Given the current state of our knowledge what is the probability that the attack on the MSF hospital was intentional?

Let me clarify two points before I try to answer this question. First, there was a sustained attack on the hospital lasting more than half an hour so we can be quite sure the building was destroyed on purpose. Nevertheless, I reserve the term “intentional” in this article to capture the case that the building was destroyed despite knowledge that it was a hospital. At this point in time it is not clear that the attack was intentional in this sense. Second, I use the term “probability” similarly to how, for example, we might discuss the probability that Argentina will win the Rugby World Cup in 2015. There is a current assessment (odds of sixteen to one against Argentina) but this can change over the next few days, e.g., if new injury information becomes available. Similarly, our view of the probability that the attack was intentional will evolve over time in response to new information. For example, relations about weaknesses in systems for protecting special coordinates from attacks may render unintentional scenarios more plausible than they may have seemed at first glance whereas a transcript of conversations between pilots and their controllers could settle the question one way or the other.

We can use the Jon Schwartz article to make a very rough estimate of the probability that the MSF strike was intentional, working by analogy with similar cases and the extent to which the evidence in these cases points toward intentionality or unintentionality.[1] To this end, I went through the links and capsule descriptions Schwartz provides and scored them according to how convinced I am that the US had attacked with knowledge that it was hitting a civilian facility. These are subjective calls that I will post soon on my personal blog to give readers a chance to second guess me. The ratings are between 0 (definitely unintentional) to 1 (definitely intentional) although I never go to either extreme since it is hard to be absolutely certain. My average over the eleven incidents suggests that the odds on “intentional”, as opposed to “unintentional”, are around 10 to 3 against. However, hospitals are special and none of the eleven attacks were on hospitals so I adjust my estimate upward to 5 to 1 against.

This is an educated guess that could be way off, not a solid number. The idea is to reason about the case at hand by thinking through analogous cases. I feel that this thought experiment has been illuminating. I learned, for example, that key US officials, Democrats and Republicans, have viewed television stations as propaganda outlets, hence, potential military targets. I also learned that air-strike targeting has sometimes slumped to Keystone Cops levels of competence, e.g., striking a Red Cross facility, conducting talks with the Red Cross on how to avoid further such mistakes and then immediately striking the exact same facility. In short, the historical record suggests that we should not instantly rule out the possibility that the MSF strike was intentional.

As with the Sally Clark case we need to examine the specifics of the case before forming a mature judgment. We cannot divine what really happened in Kunduz purely through historical extrapolation. But we also must not foreclose plausible possibilities from the outset. A proper investigation of the MSF attack needs to remain open to the possibility that it was intentional as well as to the possibility that it was a terrible unintentional mistake.


Michael Spagat is head of the economics department at Royal Holloway, University of London. He blogs at War, Numbers and Human Losses.
[1] This estimate is in the spirit of the wonderful new book Superforcasting: The Art and Science of Prediction by Phillip Tetlock and Dan Gardner. This book provides much guidance on how to make such estimates.
Please note that this is a forum for statisticians and mathematicians to critically evaluate the design and statistical methods used in studies. The subjects (products, procedures, treatments, etc.) of the studies being evaluated are neither endorsed nor rejected by Sense About Science USA. We encourage readers to use these articles as a starting point to discuss better study design and statistical analysis. While we strive for factual accuracy in these posts, they should not be considered journalistic works, but rather pieces of academic writing. 


  1. My Second Article on the Attack on the MSF Hospital in Kunduz | War, Numbers and Human Losses - […] have a new article up at about the US attack on the MSF hospital in […]
  2. Scoring US Attacks on Civilian Facilities | War, Numbers and Human Losses - […] is a follow up to my article about the attack on the MSF hospital in Kunduz that I just…
  3. US Air Strikes on Hospitals | War, Numbers and Human Losses - […] is that none of them were actually attacks on hospitals.  However, shortly after publishing my article friends made…

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