In the early Seventies practically every child
in the land received a skateboard for Christmas. Sadly but absolutely inevitably
there were a flood of accidents. Equally inevitably there followed a public outcry
for safety equipment, despite the fact that no-one had thought of calling
for it when our earlier generation used (the actually much more dangerous)
roller skates. What is easy to fail to understand is that a tiny probability
of accident, for an individual, becomes a large probability for a large
population. In other words, if three children have skateboards, the probability
of any of them having an accident is next to none. If 300,000 have them,
there will be some who have accidents. How dangerous skate boards actually
are is not affected. If instead of skateboards 300,000 children had
been given Ping-Pong balls, at least one of them would have choked on it.
Similarly, we are told it is very important
to wear a seat belt even for a very short journey since 80% of accidents
occur within 5 miles of home. Of course they do, 80% of all journeys are
within five miles of home. The probability of an individual having
an accident is tiny, with or without a seat belt.
usually given as a number between 0 and 1. For example when you flip a coin
the probability of it coming down heads is 0.5. If you flip N pennies the
probable number (n) of heads is given by n = 0.5 x N
individual probabilities can have large effects when large numbers of people
are involved. Do not rush in to change the rules without understanding that
fact. If we denote the probability of one person having an accident P1 and
take n as the probable number of people having an accident when N people
are involved then:
One sort of confusion leads to others: Seat
belts are fine, they may reduce injury when you have an accident, but the
probability of an accident depends not on the seat belt, but mainly on
the person wearing it. Tragic accidents to children in minibuses do not
occur because they are not wearing seat belts, but because their driver
(or possibly another) made a mistake. Likewise keeping count of the number
of deaths from any kind of activity is not a useful statistic. One probability
leads to an accident, another (largely unrelated) probability decides whether
someone dies in it. The latter will depend, for example, on the nature
of the car and, more simply, on the number of people in it. The number
of deaths in an individual accident is largely independent of the probability
that the accident happened.
probabilities depend on multiplying together largely independent probabilities
as well as numbers. Taking Pd as the probability of death, and P1 as the probability of an accident, we have
the probable number of deaths n:
Back in the fifties there was a campaign to
get drivers to used dipped headlights in built up areas and this was tested
in Birmingham. Sure enough there was a reduction in accidents. However
they used total figures and in fact most of the reduction occurred during
daylight hours! due to nothing in particular.
I worked one day a week in a cafe. Some weeks
we ran out of the menu special, some weeks we sold only a few.
Probabilities are often greatly affected by totally unpredictable random
events and we denote that by R:
The prediction of weather is a bit like unreliable elastic
and so-called Chaos Theory has become fashionable.
But forget about butterfly wings in the Amazon. What actually happened
was the realisation that changing the readings from weather stations by
only a tiny amount (well within the possible error in the instruments)
could lead to totally different predictions of the weather. And this effect
got worse the further forward predictions were made. Three days is now
about the limit. Think about your own life: some decision way back, whether
you did this or that subject at school, whether you went to a particular
place at a particular time (so meeting you spouse for example) made an
enormous difference to your present situation.
Probabilities also depend on the possibly enormous effect over time following
some minor or random event. We denote that by E.
This analysis has bordered on the mathematically simplistic,
even naive, which only goes to show how even more simplistic and
naive is some of the information with which we are bombarded each day,
when it takes no account of these matters. My analysis is not here to convince
you of anything other than the need
to think and think for yourself. You, dear reader, need to
realise that most information, when being used to convince
us about something, is generated not by thought but by emotion. Because
I have a very simple rule:
I assume that:
Quality of the information I am being given is inversely proportional to
the dogmatic confidence with which it is presented.