7/7 Exercise probabilities
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The story:

An anti-terrorist exercise was running at the same time as the London 7/7 bombings, and it even predicted bombings at the same stations. The chances of this happening by chance is one in 3,715,592,613,265,750,000,000,000,000,000,000,000,000 which is more than the grains of sand in the entire world.

The reality:

Sounds staggering, right? But it's also twisting the statistics very considerably. We'd recommend you read the original calculation at http://www.infowars.com/articles/London_attack/probability_drill_attack_coinciding.htm, then we can consider the many problems.

1. The accuracy of the exercises is based on this quote from Peter Power:

At half past nine this morning we were actually running an exercise for a company of over a thousand people in London based on simultaneous bombs going off precisely at the railway stations where it happened this morning

But when asked to clarify he said "Almost precisely", and

"we based on a scenario of simultaneous attacks on a underground and mainline station"

However, there was no mainline station attacked. Power may have predicted a bomb on Kings Cross mainline station, and counted this a "hit" because a bomb actually occurred close to Kings Cross underground station, but they are not the same place. He also did not predict a bomb on a bus. 

Further, in later clarifications, Power said this:

"It is confirmed that a short number of 'walk through' scenarios planed [sic] well in advance had commenced that morning for a private company in London (as part of a wider project that remains confidential) and that two scenarios related directly to terrorist bombs at the same time as the ones that actually detonated with such tragic results. One scenario in particular, was very similar to real time events".

So now it appears that Power was running several scenarios, or at least more than two. One of these was similar (but not identical) to real-world events, we would guess in predicting two out of three of the named stations. It may be that another referred to a bomb at the mainline station instead of the underground, and the fourth bomb (on the bus) was missed altogether. Not quite as precise as the headline figure would have you believe, then.

2. The calculation assumes that the attacks are randomly spread and independent of each other. Let's look at how realistic this is.

In terms of location, they mention that there are 274 Underground stations, and assume that each are likely to be hit.  There's no reason to believe this is true. Many of these stations are far outside Central London, and bombs there wouldn't cause nearly as much disruption. It makes far more sense to hit Central London stations only, and for multiple terrorists working together from a single starting point, this greatly reduces the figure. We'd say 30 to 50 stations is far more realistic.

In terms of time, the calculation mentions that each station is open for 19 hours, therefore the chance of an attack within a particular 1 hour block is 1/19. This is also implausible. Terrorists want to cause the maximum disruption and casualties, which means striking in the morning rush hour between 8 and 9am. They did this in Madrid, it's the obvious thing to do in London.

In terms of date, the calculation mentions that the stations are open 364 days a year, and assumes each day is equally likely to be hit. Wrong again. At a minimum, they would hit working days only (that's 260 in a year). If Thursdays are a particular favourite (as with Madrid) then that's reduced to 1 in 52.

Finally, the calculation assumes that each bombing is independent. So they state that the chance of one bombing occurring at the time and place it did was 1 in 9,474,920, therefore the chances of three occurring were 9,474,920 x 9,474,920 x 9,474,920 = 1 in 850,602,500,906,920,000,000. It's a big figure, but an inaccurate calculation. The whole point of these bombings is that they're coordinated, so if one happens, then the others are going to follow. The extra multiplications only serve to make an impressively large figure, and in no way affect the real probability. 

Which should be what, exactly? They've gone for the maximum figure, so let's see how it might be if we take a more conservative view.

Likely hours of attack = 1, the 8 to 9 rush hour
Likely days of attack in 1 year = 52 if we're aiming at Thursdays
Likely stations of attack = perhaps 30 in Central London
Likely period of attack = 3 years max, not the 5 they specify

Probability of an attack on one station at a particular hour is now 1 x 52 x 30 x 3 = 1 in 4,680

Probability of 3 attacks at named stations = 4,680 x 29 x 28 = 1 in 3,800,160

...and in fact that could be even less if we decided it's likely that the attacks wouldn't occur at adjacent stations...

4,680 x 27 x 25 = 1 in 3,159,000

Which is a little big less than 1 in 850,602,500,906,920,000,000, although the problems don't stop there.

3. Even our figure of 1 in 3,159,000 seems high, but that's because we're calculating the chance of the bombings occurring at that particular time and date. Again, that's irrelevant, just a trick used to boost the figures. The probability we need to be finding out is given a terrorist attack at any time (regardless of whether it occurred on the 7th of July, 8th, 9th or any other time), how likely is it that a matching antiterrorist exercise would occur at the same time? 

Think about it that way and we have to ask other relevant question, that the original article entirely ignores. Like, how many antiterrorist exercises are going on? If Peter Power has one of these a week, for instance, then the chances of a hit on the same day are 1/5. And it's likely they'll be at the same time, too: between 8 and 9am for a terrorist hit, with the exercise beginning just after 9am once everyone's arrived for work.

4. There's still the probability of predicting the stations, of course, but what counts as an accurate prediction? That's not clear.

The first bomb exploded in the tunnel between Liverpool Street and Aldgate, for instance ( http://en.wikipedia.org/wiki/7_July_2005_London_bombings ). Some media reports described it as a bomb at Liverpool street ( http://news.bbc.co.uk/2/hi/uk_news/4662045.stm ), a Government statement said it happened at Aldgate ( http://www.direct.gov.uk/Nl1/Newsroom/PublicSafety/PublicSafetyArticles/fs/en?CONTENT_ID=10020593&chk=1wl1uB ).

Another bomb exploded about half way between Russell Square and Kings Cross ( http://news.bbc.co.uk/1/shared/spl/hi/uk/05/london_blasts/html/russell_sq.stm ). Again, Power is getting two chances at a hit (so he could name any two of four Central London stations and say he got it right).

It appears Power may also have accepted as a hit, his prediction of the bombing of a mainline station, when the bomb actually hit an underground station with the same name.

What does this mean? Although only three bombs went off on the underground, we've multiple chances of naming them "correctly". Liverpool Street Underground, Liverpool Street Mainline, Aldgate, Russell Square, Kings Cross Underground, Kings Cross Mainline, Edgware Road and possible Paddington Underground and Paddington Mainline may all be close enough to be scored as hits.

Factor in an extra free guess (Power didn't predict the fourth bomb, so either he only predicted 3 events or his fourth guess was incorrect), and the fact that he had more than two scenarios, and that he may only have scored two hits on the best of them, and this is beginning to look not so improbable at all.

Still not sure? Do the math. If we accept that 30 mainline stations are most likely to be bombed, and 7 of these names count as a successful prediction, then there's a 7/30 chance of a successful prediction. That's almost one in 4, so make 4 guesses and you'll probably have at least one hit. Make several sets of guesses, and it's not at all inconceivable that you'll have two, or even three. 

Of course some will argue with our assumptions. Maybe it's not 30 stations, they might say -- you should say 40, 50, 60 or more. But that's not the point. What we're trying to say is the original 1 in 3,715,592,613,265,750,000,000,000,000,000,000,000,000 probability figure is a gross distortion. The reality is very different, and the chances of these two events coinciding certainly aren't so high as to prove conspiracy.

See also a special report from UK TV station Channel 4 at http://www.channel4.com/news/special-reports/special-reports-storypage.jsp?id=372

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