We've been having an argument in the office and I'd like you to settle it. On their side is sweet reason and a sense of proportion. On my side? I think the maths is on my side.

The question is this - how seriously should we take the latest fiasco at the Home Office?

I'll put my basic case quite briefly. My faith in airport security has never been the same since I noticed that the man confiscating the shaving foam in my hand luggage (while leaving me with the razor) had the word HATE tatooed on his knuckles. But, call me naive if you like, I did imagine that security staff cleared by the Government's Security Industry Authority had been properly checked. Otherwise, what's the point of all that queueing and “Would you mind if I looked in your bag, Sir”?

Now it turns out that the checks are a sham. The problem is not, as admitted by the Government, that 5,000 of these staff are illegal immigrants. Or that such people have been guarding airports, Scotland Yard and the Prime Minister's car. No, the problem is that if 5,000 illegal immigrants have been adjudged “fit and proper” people, it suggests we don't really know anything about any of those that have been cleared. It calls into question the whole system.

Bad. But worse is that when the Home Secretary and her ministers discovered the problem, they elected not to tell anyone. This was a deliberate strategy, the press office having concluded (imagine the meetings and then the moment of inspiration) that “it would not be presented as a positive story by the media”. It seems almost certain that, if Gordon Brown had called an autumn election, the Home Secretary would have sought to conceal the story throughout the campaign.

So my side of the argument is this: if not actually a resignation issue, this comes close. The failure is catastrophic, the decision to conceal it an outrage. It ought to shake the careers of everyone involved.

The counter-case is that I am being hysterical. The Home Office is huge, its bureaucracy sprawling. Failures happen all the time, they are bound to, and you can't expect the Home Secretary and her ministers to take the fall every time there's a bad news story. As for the press office business, yes it's incompetent, even comical, but lighten up. Don't turn everything into a story about the media. The whole saga came out in the end, didn't it?

What we might call the Michael Winner approach to scandals (“Calm down, dear”) extends beyond this case. Many thought it overheated to call for the Metropolitan Police chief, Sir Ian Blair, to stand down over one tragic incident. In The Sunday Times, for instance, Simon Jenkins decried the way accountability has become a “random bloodsport”. And certainly the way one minister is forced out while another survives a worse error is often hard to understand, let alone justify.

So I can see the argument for getting it all in proportion. It always exerts a strong pull on me. Perhaps it would even win me over. If it wasn't for the maths.

In a series of books on the financial markets, Nassim Nicholas Taleb bemoans the failure of dealers to take into account not simply the probability of an event, but also its consequences. As a result they underestimate the importance of extremely unlikely but disastrous incidents. In order to determine how seriously to take a future eventuality you have to multiply the small probability of the event occurring, by its vast impact if it does occur.

A similar maths is at work with scandals. First, turn on its head the argument that we shouldn't overreact to incidents like the security checks because they are routine. Instead, understand that this is precisely why we do need to overreact. If the Home Office hardly ever made stupid mistakes, we wouldn't need to worry about it. It is the unexceptional nature of the incompetence that makes it so dreadful.

Next, realise that almost all these mistakes are hidden from view. Most aren't noticed in the first place, the incompetence carrying on quietly. Then there are those where the cover-up works and the incident is never revealed. Risible e-mails are guffawed at in private. Half the time even the minister isn't told.

How can you police the competence of a chronically incompetent organisation where the probability of discovery of any single error is very low? Answer: make the consequence of discovery very high. That way you provide the correct incentive to staff and ministers to be competent. They will multiply the probability by the consequence - knowing that it is unlikely that their mistakes will be unearthed, but that if they are unearthed they cannot expect the press and public to be all sweet reason.

Simon Jenkins is right: accountability is a “random bloodsport”. And so it should be. Press frenzy may not be very edifying, and its randomness hard to take, but these are essential features of a properly functioning democracy.

Isolate the horrendous error over the security checks and the Keystone kops/Nixon White House (delete according to taste) antics of the press office and conclude that it isn't a capital offence. But look at this incident as one of many and it appears different.

In a rare, rare moment we have caught a glimpse of what really goes on in the Home Office. We have discovered that they gave government security clearance to illegal immigrants, let them guard the Prime Minister and then covered up the story. Imagine the lesson learnt by politicians and the Civil Service if even a scandal like this, upon reaching the newspapers, turns out to have little or no consequence.

My colleagues may be reasonable. But sometimes a little overreaction is in order.

daniel.finkelstein@thetimes.co.uk

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