I see a lot of
headline-grabbing warnings in the media about how activity x, y or z is a huge
danger to a, b or c. They bandy figures around like if you do x you'll be
80% more likely to get this type of cancer, or y increases the risk of a heart
attack in men by 70%, or z makes a more than 5 degree temperature more likely
by 75%.
When stated like that,
activities x, y and z can quite easily cause alarm - and they often do, whether
it's how many cups of tea you drink a day, how often you have a fried
breakfast, how many cars are on the road or whether you smoke in a house with
children - someone has got something to say about it and some stats to throw
at it.
But those figures mean
very little unless you know the base probabilities to begin with. For example,
suppose drinking an extra two cups of tea a day (from, say, 4 to 6 cups)
increases the probability of your getting prostate cancer by 40%. On first
inspection it may sound advisable to stick to the 4 cups of tea a day.
But suppose drinking 4
cups of tea a day only makes your chance of getting prostate cancer 15% - a 40%
increase in probability from 4 cups a day to 6 cups a day only adds another 6%
to your chance, which is still only a 21% chance. You may well feel that all
those extra cups of tea over your lifetime is worth a 6% increase in
probability. However, if you just saw the headline "2 extra cups of tea
increases the chance of prostate cancer by a whopping 40%" then taken at
face value it may put you off tea for life.
We routinely hear claims
of the kind that eating two rashers of bacon a day raises the risk of bowel
cancer by 18%. But without a base rate (how common is bowel cancer?) this
information is not very useful. As it happens, in the UK
Here's another example. Suppose
40 million of the UK
But, of course, that's
totally the wrong way to think about it, because we need to know the base rate
- that is, the number of people who trusted the media in the first year.
Apparently the actual figure is that only 6 out of every 100 people say they
trust the media, so an 18% increase the year after is only an extra 1 person in
every 100 now trusting the media.
Numbers are misleading
- they can shock in large quantities, but that often skews the real picture.
When a few years ago Vince Cable projected that our joining the Euro would
increase our GDP by a few billion pounds, lots of Liberal Democrats got
excited, and many pressed for us to join.
What should
have been obvious is that that GDP figure is spread over the entire population
of Britain, and doesn't add up to much at an individual level (for example, 4
billion divided by 63 million works out at just over £60 each). Would you want
to lose the pound sterling for an extra sixty quid in your pocket? (that was
then, of course - now almost every Brit is glad we didn't touch the Euro with a
barge pole).
Yet another example. Imagine
that there is a new illness discovered, colloquially called 'MXDA', the
symptoms of which are swollen hands and occasionally swollen feet (with neither
causing the other, and either can occur independently of the other). With MXDA
swollen hands occur in 99/100 people diagnosed with it. Swollen feet occur in
only 1/100 people diagnosed with it. Consider this question: which of the two
following statements is most probable:
A) George contracted MXDA
and had swollen feet
B) George contracted MXDA and
had swollen feet and swollen hands
The vast majority of
people answer B, even though the answer is obviously A. It's fairly evident
that even though George has swollen feet, there is still a 1 in 100 chance that
he does not have swollen hands, and given that the two probabilities are
independent, it is impossible that B is more probable than A. Belief to the
contrary is known in philosophy as the 'conjunction fallacy'.
Understanding base rates,
probabilities and logical thinking can help us in everyday life decisions too.
Take insuring your household products (a subject I once blogged about here)
Using similar logic to the above thinking, then in terms of probability when it
comes to insurance of household products the cards are stacked in the favour of
the insurer not the insured. It should be fairly evident why: insurers must
cover the cost of pay outs and the administration costs to sell insurance, so
insurance must be a winning hand for the insurer overall (ditto casino owners, bookmakers, amusement arcade owners, and so on - the fact that they are in business at their customers' expense tells us all we need to know).
Should someone who has
spent £20,000 on a conservatory spend an additional £30 to insure against it
being damaged by the weather? The obvious answer seems like yes, but it depends
on the 'base rate' - the odds of the conservatory being damaged by the weather.
If the odds of it being damaged are 1000/1 he'll be £10 down on the deal. Maybe
it's still worth it, but what about £3,000 insurance against a 1000/1 chance
that a £2 million conservatory will be damaged? It's the same as before in
terms of odds and ratio, but our man may think an extra £3000 could be better
spent elsewhere.
One final point, when it
comes to perceived rationality things aren't always as they seem - sometimes
it's important to think a bit further outside the box. For example, generally
it is thought that people who understand probability won't buy a lottery ticket*
because they know the vanishingly small chance of winning the jackpot pretty
much makes any lottery ticket purchase tantamount to a waste of money.
That might be true if
those gambling odds were all there is to the purchase, but there are other
factors that might make a ticket purchase worthwhile, just as there are other
reasons why going for a night out at the casino is not necessarily irrational
despite the probability being in the casino owner's favour. In buying a lottery
ticket you might also be buying the dream and the excitement, which may well be
worth the value, particularly if you enjoy the whole TV show that goes with the
lottery experience. And in case you're wondering, I don't buy a lottery ticket,
I'm not an idiot! Haha! Just kidding!
* I remember reading about the high number of lottery
winners who squander their fortune, some of whom even go bankrupt within a few
years. It could be that the paradox of lottery players is that if they are
willing to spend a few pounds each week on lottery tickets in the first place
they are not likely to be the kind of person who optimises their spending
commensurate with their budget. Therefore, one would expect that a high number
of lottery winners had proclivities for profligacy. "Proclivities for profligacy" - That's one hell of a
statement.
