In "General Betting" somebody asked whether these odds stood up to scrutiny:
Of course they don't. I answered as follows:
"They say you can prove anything with statistics, but to be honest you can badly attempt to disprove more through a poor understanding of probabilities...
"Coincidence" and in particular "amazing coincidence" is normally better understood as a manifestation of confirmation bias. Our brains are hard-wired to spot patterns and to remark at as being unusual when patterns we weren't expecting emerge. We see patterns where there aren't any. We scream about patterns we "see" that are nonsense, because that's evolution and defence mechanisms at work. Confirmation bias explains a lot about conspiracy theories: http://en.wikipedia.org/wiki/Confirmation_bias
So where is it at play in the example you posted? The odds being chosen are selective and then being incorrectly multiplied up as wholly independent events, which is flawed, and this makes it "seem" more unlikely.
For example, the figures stated imply that all hours of the day have both an equal chance of being subject to a terror attack and equal chance of having a drill being run. That's flawed: terrorist attacks will be targeted at rush hours when the network is at its busiest, and drills will start near the beginning of the day in order to ease securing personnel at normal day rates, etc.
He's also assumed that terrorist attacks will only happen once every five years and drills will also happen once every five years. At the beginning of 2005 I would imagine most security service staff would have not even given you even money on an attack within 12 months. I expect drills to practice for terrorist attacks probably happen more than 2-3 times per year.
Odds of a morning being subject to a terrorist attack in 2005? 1 in 364. Odds of a morning in 2005 being start of a drill? 3 in 364. Odds of them coinciding? About 500/1.
So his figures are vastly inflated. Worse he then takes the figures and cubes them because there were 3 attacks and drills being run in 3 stations. This only makes sense if all the attacks and all the drills were independent of each other and in no way connected. Of course they weren't: it was a single terrorist attack, with 3 targets (and so no multiplication or addition is necessary), and it was a single drill being run by London authorities, not three drills being run by three separate groups that happened to be on the same day.
True odds then are way shorter. Depending on what (fairer) numbers you use, and you avoid multiplying them, it comes out around 250-500 to 1. The vast majority of the time it isn't going to happen, but it's going to happen way, way more than those figures predict.
For all we know, the attackers were aware of the drill, and wanted to make use of the confusion having an attack on the same day would produce in the emergency services. If that was the case, the odds were nearer 1.01...
ALWAYS question probabilities that are cited by conspiracy theorists, especially if they're multiplying them up to make them impressively large. HTH..."