Elastic and Other Probabilities

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Clown Skiing

Start with this fact:

If a large number of people make use of something, or take part in something: some will suffer a failure of some kind. Elastic will break, some skiers will have accidents.

Elastic lasts better now we wash it at 40 degs C,
rather than boil it at 100,
but that's another story

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.

Probabilities are 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

Point 1 Tiny 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:

n = P1 x N

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.

So Point 2: Final 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:

n = P1 x N x Pd

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.

Point 3: Final Probabilities are often greatly affected by totally unpredictable random events and we denote that by R:

n= P1 x N x Pd  x R

In many situations the effect of random events evens out over time. For example we still bought the same number of specials each week in the cafe. We might compare the variations in the number sold to elastic, which returns more or less to its average size. Sometimes, however, elastic breaks, especially when stretched too far. Or this may occur as the result of one small stretch too many.

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.

Point 4. Final Probabilities also depend on the possibly enormous effect over time following some minor or random event. We denote that by E.

P = P1 x N x Pd x R x 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 of that I have a very simple rule:

I assume that: The Quality of the information I am being given is inversely proportional to the dogmatic confidence with which it is presented.

That applies to everything from statistics on smoking to interpretation of the Bible

But there is another point, if the quantities R and E (above) are all important, And we have no control over them, where does that leave our understanding of morality and indeed of God. A friend who I visit in prison suggests that the present situation between him and his guards is largely a matter of chance. Their present roles could easily have been reversed by some random event in the past. To answer that question is to obtain the answer to:
The Meaning of Life

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