The Blue Bus paradox

How probability-based decisions can lead to legal injustice

Sometime around 1am on 6^th February 1941, Betty Smith was driving down Main Street of Winthrop, Massachusetts when she spotted a ‘great big, long, wide affair’ heading directly towards her at speed. To avoid being hit by the oncoming bus, she veered to the right, crashing into a parked car.

Although Smith couldn’t be sure which specific bus she’d seen, only one bus company – Rapid Transit – had a licence to operate on the street. What’s more, their buses that night had been scheduled to leave Winthrop Highlands, less than a mile from Main Street, at 12.10am, 12.45am, and 1.15am.

Given these facts, Smith took Rapid Transit to court. Because it was a civil case, it would come down to ‘preponderance of the evidence’. In other words, was it more likely than not that a Rapid Transit bus had caused her to crash into that parked car?

At first glance, it might seem like the evidence was in Smith’s favour. Rapid Transit was the only company with a licence to run a service on that street, and they had scheduled buses around the time of the accident. Surely this would mean there was at least a 50% probability that a Rapid Transit bus was responsible?

Yet the judge in the case ended up ruling against Smith. ‘While the defendant had the sole franchise for operating a bus line on Main Street, Winthrop, this did not preclude private or chartered buses from using this street,’ he noted. ‘The bus in question could very well have been one operated by someone other than the defendant.’ In his view, this meant there was too much doubt about who was responsible. ‘The most that can be said of the evidence in the instant case is that perhaps the mathematical chances somewhat favor the proposition that a bus of the defendant caused the accident,’ he said. ‘This was not enough.’

The Blue Bus paradox

Why was it not enough to look only at the chances a bus was owned by a particular company? The Smith case would inspire a long-standing legal puzzle, known as the ‘Blue Bus paradox’. Imagine a pedestrian gets hit by a bus and injured, but it was dark and so they can’t recall the colour of the bus responsible. Regardless, the pedestrian sues the Blue Bus Company, because they own 75% of the buses in the town, which means chances are it was one of their buses. Based on preponderance of the evidence, should the pedestrian win?

When faced with probability puzzles, it can be helpful to consider an extreme example to get more intuition. Take the related ‘gatecrashers problem’, first posed by philosopher L. Jonathan Cohen:

Consider, for example, a case in which it is common ground that 499 people paid for admission to a rodeo, and that 1,000 are counted in the seats, of whom A is one. Suppose no tickets were issued and there can be no testimony as to whether A paid for admission or climbed over the fence. So by any plausible criterion of mathematical probability there is a 0.501 probability, on the admitted facts, that he did not pay.

Mathematically, it seems that the rodeo organiser should be able to recover the cost of admission from person A, because there’s a more than 50% probability they hopped the fence. But by this logic, the organiser could claim successfully against all 1000 people who attended, even though 499 had paid.

We have a similar paradox with blue bus case: 75% is more than 50%, so at first glance it may be tempting to find the Blue Bus Company liable. But now imagine this same injury-by-unknown-bus situation crops up repeatedly. In every case, we could make same argument, which would mean the company losing 100% of the cases, even though they only own 75% of the buses. In statistical terms, this creates a bias where the Blue Bus Company is systematically over-penalized, simply because they are more common – not because they are actually responsible in each specific case.

The paradox illustrates the dangers of using broad probability arguments for individual legal cases. I’ve written previously about the challenges of predicting individual outcomes from population averages, and we see a similar issue when it comes to the blue bus paradox. In reality, courts need to look at the direct evidence relating to the case. Were there any witnesses? Could another vehicle have been there? What other evidence is available that links the bus company to the incident in question? ‘

‘Preponderance of the evidence’ does not just mean that one explanation (i.e. a blue bus) is more likely than another specific explanation (i.e. a different bus bus). To meet the threshold, the claim of a blue bus being responsible must be more likely than all other competing explanations combined. What if the pedestrian was hit by a private coach mistaken for a bus? What if there was no bus involved at all (e.g. a hit-and-run by a truck)? What if the pedestrian fell due to an unrelated reason (e.g. a pothole) and blamed a bus?

The Blue Bus paradox is a useful, if counter-intuitive, reminder that justice isn’t about broad statistics; it’s about specific evidence. If we were to let baseline probabilities dictate verdicts, we’d risk punishing the likely without evidence – instead of proving who was guilty.

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If you’re interested in probability and proof in the courtroom, you may be interested in my new book Proof: The Uncertain Science of Certainty, which is out in the UK this week and available to pre-order now.