# Probability

## Good use, and misuse, of probabilities

A probability is a mathematical ratio that expresses the likelihood
that an event will happen and how often (frequency).
It is used for prediction and decision making.

Probabilities should be used instead of guesswork when they are
relevant.
But statistical distributions can mislead by giving obsolete or
insufficient data that do not match new or rare situations.

What are the chances that it will happen?
Or not happen?
Hoping it does!
Or not!

## Definition

A probability is a mathematical ratio, usually expressed as a
percentage, that shows the expected likelihood and predictable
frequency of an event.

To put it simple, it is
how many times - in average - it will
happen
if there are 100 draws (or equivalent events in real life: ask any
insurer how many people get hit every year by a falling flower pot, nothing
theoretical here, I once had a near-miss).
To take a classical example, when throwing a coin (not the warped
one you use to bet at the bar), there will be in average 50 cases in
which heads come out.
The probability ratio is therefore 50% for heads.

Of course, you will not see it if you make only 100 draws,
You might even get the impression that heads are more frequent
than tails or the other way round.
This defect is known as "the law of small numbers", which
probability based  onthe "law of large numbers".
Thus if, on a rainy Sunday, in which you have no other amusement
in sight, you make 100 hundred times 100 hundred draws, you will
come very close in average to the 50/50 ration.

## How to find the probability of an event

### The role of statistics... if they are relevant

This ratio that gives the odds most often is found by observing statistical
distributions (breakdowns in series of numbers, ordered from the
smallest to the largest) that show the past frequencies of events.

Past frequencies? Here start the problem.
There is a range of cases in which it is difficult to find reliable past ratios:
• The series / sample might be too small / too short
We have here the above mentioned caricatural "law of small
numbers", that can give an exactly opposite image of the real
situation or evolution, or totally miss or exagerate the weight
of some possible "rare" events.
• Various new situations are without past equivalent
Sorry, no past statistics, first time we produce beauty gel,
our previous specialty was furniture varnishes!
• Some distributions are erratic / chaotic / distorted and disrupted.
They do not allow to find stable and precise probabilities.
• Others shows frequencies that seem more stable
but which cannot be translated into a mathematical
law / curve
. Fuzzy shaped animals!
• Others seems so stable, clear-cut and regular
that they can be associated to a law of randomness.
They are so beautiful that some Masters of stochastic (an
area of maths devoted to probabilities) are sometimes
tempted to generalize them
in their applications,
notably in finance.

### As a classical example of the fourth case,

The famous "normal" law, found by Gauss and Laplace,
shows on a chart a bell shaped curve.

* A peak of occurrences situated around the mean (average number
of occurrences, top of the bell).

* A symmetry on both sides of that mean.

* A precise measure of the deviations from the mean. We use for that
purpose

- Either the standard deviation
It includes about two thirds (more precisely 68%) of the cases,
in the two central "quartiles" of the series

- Or the variance
It is the sum of the squares of all deviations divided by the
number of variations

The smaller are those indicators of variation, the narrower is the
statistical  "dispersion" of data.

=> To illustrate this
, in a population of male adults,
* Few will be under 1.6 meter high or over 1.9 meter high

* The mean would be 1.75 meter and there will be a
concentration in the 1.7 - 1.8 meter range.

* If two thirds of the value are located in a 1.66 - 1.84 range,
in a symmetric way around the mean,
=> the standard deviation would be
(1.84 - 1,75) = (1,75 - 1.66) = 9 cm
 Gaussian distribution X X X X X X X X X X X X X X X X X Horizontal scale = values (i.e. people sizes) Vertical scale = frequencies (number of times each value occur)

Careful:

This standard deviation (and the normal law) is valid and can help
calculations only if the bell curve is not too
distorted, with
asymmetries, clusters, flat curve, disruptions, too narrow "tower"
in the middle, too thick
tails (extreme values) showing many
exceptional events...

## Practical applications

Statistics are used notably in situations and projects in which risk (and
opportunities) can be measured or at least estimated.

They are important tools in many areas:
* For insurance activities
(frequencies and sizes of accidents...), a quite common example.

* In finance,
notably in calculations based on market price and return "volatility",

* In economics, management and business generally.

* And of course in many other areas:
medical activities, weather forecasts, technological projects, oil
drilling, public and private security, geostrategy and much more...
More generally, probabilities are used as prediction tool in various
decision making situations.
For example, somebody who buys and sells garments needs an
idea of how many S-sized and XXL-sized potential clients will
enter its shop, so as to adjust its inventories.

## Should we trust statistical probabilities?

### The danger of ignoring them or ...inventing them

To neglect statistical probabilities is quite dangerous.
It can lead to a distorted vision of the risk taken when
making a
decision.

This is a mental bias called the base rate fallacy. In other words to
rely on a subjective impression that does not fit realities.

Practically, people tend to overestimate low probabilities and
to underestimate high ones
.
In their mind 10 % becomes 1/3 and 90% becomes 2/3.

### The danger of trusting them without precaution

Look at the horse's statistical teeth!

Historical series, are far from fully
reliable, at least in
situations of upheaval as we see them nowadays.

Thus, however useful they are in many situations, statistics sometime
.
To trust them too much without checking their relevancy is another
mental bias called the numeracy bias.
Our friend the shopkeeper we talked about might find insufficient
the professional statistics and useful to sneak at night into its
competitors premises to see which sizes are missing and which
ones are overflowing their inventories.
Not that I advise it!
Last but not least, many statistics cover a too short period and

* not only give a
false perceptions of the frequency of ordinary events

* but also
usually do not show rare events (the 100 year storm, the
"black swan") or on the contrary exagerate their frequency.

=>
Some of those exceptional events
(be they a tsunami or a liquidity crisis)
might cause extreme havoc
.

To ignore their possibility and not to protect against them might bring
tragedies.

What is deemed to happen only once in a billion years according to
the normal law of distribution might in fact have a real probability
to occur in the next five years.

## Alternative tools

In complex dynamical systems, statistical probabilities cannot always
be applied. They might be unavailable or irrelevant (something seen
in many new situations).

In such cases, risk cannot be measured objectively. We have then a
case  of  uncertainty (a non measurable risk) in which we
have to:
• Use guesswork (subjective probabilities)
and build tentative scenarios (including extreme case
scenarios),
• Adjust them after every new event
(Here we have conditional or Bayesian probabilities).

Another alternative method is fuzzy logic, that we might label "vague
probabilities".
It was devised for situations that lack precisions, which is often the case
in dynamical (= evolving) systems.

## What about economic models?

One of the lessons to draw is that economic models and projections,
teeming with mathematical equations, which are currently highly
criticized if not derided,
should be made flexible on the basis of
scenarios
taking into account "surprises" such as:
• Not only quasi mechanical extreme events / disruptions
they are proper to most dynamical systems,
• But also the "soft" elements
such as the possible changes of attitudes by players.

## Source and further readings

More details on probability, risk, uncertainty and related
biases in the
Behavioral finance glossary

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