Probability

The notion of probability was possibly first developed by Gerolamo Cardano (1501-1576), an Italian Renaissance mathematician, physician, astrologer and gambler. Cardano was notoriously short of money and kept himself solvent by being an accomplished gambler and chess player. His book about games of chance, Liber de ludo aleae ("Book on Games of Chance") , written in 1526, contains the first systematic treatment of probability, as well as a section on effective cheating methods.
 (from Wikipedia).

Contents

• Frequentistic probabilities
• Subjective Probabilities
• Risk
• Combining Probabilities
• Conditional probabilities
• Prior and Posterior Probabilities

Subjective probabilities

A typical question asked is: "How do you rate the chance of success?". Terms as "very likely", "probable", "unlikely" and many others are used to give a subjective, intuitive estimate of a probability. The alternative is to assemble data and count frequencies of previous events, or some more sophisticated statistical method to come to a probability as a number between 0 and 1. If we want to have prospect appraisal results, (volumes and chances) in a form suitable as input in, say, a Monte Carlo economic analysis program, numbers are more suitable than words. Therefore the problem of estimation often is estimating a value X, where X may refer to a probability or to a numeric value of a variable. We may ask "What is the chance of having a sourcerock?" or "What is the thickness of the sourcerock". In the first case we estimate a number between 0 and 1, in the other a thickness in feet or meter. For the estimation of numbers/quantities I recommend the experiment on estimating sales of movie ticketts by Delfiner (2008).
Many psychlogists have done research to see how the human mind estimates a probability (Stheeman et al., 1993 and also McLane et al., 2008). Experimenting with groups of persons, a correlation could be made between a list of probability terms and their numeric equivalents, as shown below. The probability scale represents the "perceived probability" that the subjects thought corresponded to their linguistic terms.

From this and many other similar experiments it is clear that such subjective terms can be translated meaningfully into numerical probabilities, although not with much precision. Not surprisingly, the "double negation" statements as "not uncommon" are the vaguest of the lot. It is intriguing to see that no clear statement is available for the 50% probability. However a "likely" chance, or a "fifty/fifty" chance is a common term for such a situation. In my experience, it is best to have a set of predefined terms, such as likely, very likely, certain and impossible, arranged in a scale of increasing confidence. A.Coffinier ( Shell, 1975, who devised this scale) suggested that weights could be assigned that increase with a factor 2 along this scale, except for the extreme ends. That would look like this:

Degree of Confidence	Weight factor
Impossible	0
Extremely unlikely	1
Very unlikely	2
Unlikely	4
Likely	8
Very likely	16
Extremely likely	32
Certain	Infinite

An experiments with 30 subjects allowed to estimate the relative likelihoods of successive steps in this scale of degrees of confidence". Interestingly, subjects would switch to the next higher term, or the next lower, when the (true) likelihood changed with a factor 2. This result is consistent with biological research on stimuli, where, for instance, increased pressure on the skin will trigger a response only when the pressure is doubled, and therefore appears to justify the above weights.

The mechanics of using this scheme is to calculate the expectation from the scores assigned to the 8 classes, using these weights. In the case of a probability we would end up with a single estimate of the probability, but one which should better represent a subject's thinking than asking immediately for this single number. Note that the mechanism should return zero if there is any score assigned to Impossible, or 1, if any score for Certain. These extremes would simply override any intermediate statements. Although the method might appeal to some because of the use of common words as a starting point, it is not proven that it has any advantage over using a simpler approach.
In practice, the above scheme does not deviate significantly from one in which we let a geologist start from a numeric scheme of estimating a histogram. First the histogram classes are fixed. For a probability one could use e.g. 10 classes of width 0.1. The the user should assign scores subjectively to these classes. Their scores would be the weights to be assigned to the class-midpoint. Here the scores are percentages, the sum of which should be 100.
This method is the "user-defined histogram" or "UDF". The class boundaries are always chosen to be relevant to the variable to be estimated.

Here is an example:

This statement would give a probability of 0.365 by calculating the weighted mean value of this histogram (expectation)
([5*0.15 + 15*0.25 + 50*0.35 + 20*.45 + 10*.55] / 100 = 0.365).

There are some drawbacks to this approach. In the example it is impossible to get any probability less than 0.05, or greater than 0.95. This is quite unrealistic. However, how good are we to estimate extreme chances? My experiments with a a fairly large number of subjects showed that, when asked to give confidence intervals of some variable to be estimated, these were most of the time too narrow. That is: too many true values fell outside their 95% confidence interval (it should be only 5%). Even worse are results for the 99% confidence interval. This is somewhat in contrast with the extremes of the probability range in the figure given above: People are giving a much narrower range for probabilities at the extreme ends, which means there is consensus about what a particular word means. However, consensus does not mean we are right!

For estimating a value, such as an average porosity, the UDF appears straightforward. But what about estimating a probability as we did in the above example? Does it make sense to be uncertain about a probability? To answer that question we have to return a moment to the frequentistic probability. If the probability is calculated as the ratio of a number of particular events over all events in a limited sample, we can use statistical rules to establish how uncertain our probability estimate is. In the geological practice, we rarely have such data to lean on. Some statisticians would say that we have no "probability" at all if we estimate it subjectively (or if you wish intuitively), based on some experience database in your brain. They would call it "degree of confidence" a term that we already used above for the scores. In practice, such distinction is not too important. This leaves us with the fact, like it or not, that there is usually uncertainty about an estimated probability (P) or about a degree of confidence.

In further use of these probabilities it may matter whether we use a single number for P or a distribution ("histogram") of possible P values.

Risk

My definition of the word "risk" is probability of failure. Hence it is the complement of the probability of success ( Risk = 1 - POS). Risk is the upper part on the vertical axis of the expectation curve, down to where the curve turns off to the right.
In prospect appraisal risk is used in the sense of "straight risk", the probability that a certain essential ingredient of the petroleum system is absent. For instance the risk that there is no viable reservoir. The straight risk is usually a subjective point estimate of a probability.

Prior and Posterior Probabilities

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