Drink Driving Test

[Mince]

Imagine that police officers are given breathalysers to test for drunkenness. These breathalysers never fail to detect a truly drunk person, and they also have a 95% chance of correctly detecting that someone is sober (wrongly stating that they are drunk for the other 5%).

Now imagine that one in a thousand drivers is driving drunk.

A police officer stops a driver at random and the test indicates that he is drunk.

Assuming you know nothing else about the driver, what is the percentage chance that this person is actually drunk?

[Tarquin Thunderthighs lll]

Shickshty sheven. Hic!

[Mince]

Sigh. 

The answer is actually 2% chance that the person is drunk.

[Diane CBPFC]

I would have said 95% for this person who tested positive. 

[Tarquin Thunderthighs lll]

Sigh. 

The answer is actually 2% chance that the person is drunk.

Why would anyone ever actually need to know this unless as part of a maths exam? I’ve not needed to do a maths exam for 45 years.

[Mince]

Here's the reason why:

Imagine 1000 drivers are tested.

Of those, one will be drunk, and the test will correctly indicate that he is drunk.

For the other 999 drivers, 95% (949) of them will be correctly flagged up by the test as sober, and the other 5% (50) will be wrongly flagged as drunk.

So out of the 1000 drivers, 51 are flagged as drunk, of which only one is actually drunk.

1 out of 51 is 1.96% which is about 2%.


It's called the Base Rate Fallacy. Because the vast majority of those tested are sober, the test is much more likely to throw up a false positive than a true positive.

[Diane CBPFC]

But the person pulled over tested positive and there is only a 5% chance of a false positive so 95% chance they are drunk.

[Mince]

But it's much much more likely that the man is sober and that the test wrongly states that he is drunk than the man is truly drunk.

[Tarquin Thunderthighs lll]

Ah, but they back it up with a blood test, a walk-the-straight-line test, and a couple of stiff double brandies to calm the drivers' nerves.

[Diane CBPFC]

lol

[Mince]

Another way to see the problem with the test is just to assume that no one is drunk. For every 100 people tested, 5 will be flagged as drunk even though none of them are. The test is useless only because base rate of drunkenness is so low, and so it throws up too many false positives to identify which one is the true positive.

This is why it's called the Base Rate Fallacy.