"It is remarkable that a science which began with the consideration of games of chance should have become the most important object of human knowledge."

Pierre-Simon Laplace

History

Probability theory originally grew out of problems encountered by seventeenth-century gamblers. It has since developed into one of the most respected and useful branches of mathematics with applications in many different industries. Perhaps what makes probability theory most valuable is that it can be used to determine the expected outcome in any situation—from the chances that a plane will crash to the probability that a person will win the lottery.  In other words, probability theory, followed by 1) the invention of numbers and 2) the development of mathematics, may be considered the 3rd greatest revolution to effect how people think and relate to the world in which they live.  Never before has their been a tool that can reliably predict the future with numbers.   This remarkable fact is best illustrated with a simple pair of dice.  The probability of rolling snake eyes (two ones) with a pair of dice in the future is 1 in 36.   It is not 1 out of 35, or 1 out of 37.  What is remarkable about this fact is that it represents a reliable prediction about the future of an event expressed in terms of numbers. 

 The Unfinished Game

"Devlin uses the critical letter from Pascal to Fermat in which he discusses "the problem of points"-that is, how to determine the probable outcome of a game of chance-as a framework for a history of probability theory and risk management, fields which now dominate our social, political and financial lives."        Amazon.Com

Basic Concept

Probability theory is a branch of mathematics concerned with determining the likelihood that a given event will occur. This likelihood is determined by dividing the number of selected events by the number of total events possible.   Probability is always represented as a fraction.  For example, the number of times a "1 dot" turns up when a die is rolled is 1 out 6, or ⅙.  The number of times a head will turn up when a penny is flipped is 1 out of 2, or ½. Thus the probability of any event always lies somewhere between 0 and 1. In this range, a probability of 0 means that there is no likelihood at all of the given event's occurring.  A probability of 1 means that the given event is certain to occur.

"All knowledge degenerates into probability."

David Hume

The Three Major Methods Used to Determine Probability Values:

Subjective Probability
Prior Probability (Classic or Theoretical)
Empirical Probability (Frequentist)

Subjective Probability

Subjective probability is a probability value based on an individual's best available knowledge and personal judgment about how likely a particular event is to occur. It is not based on any formal calculations but is a reasonable assessment by a knowledgeable person that reflect the subject’s opinions and past experience. 

Like all probabilities, a subjective probability is conventionally expressed on a scale from 0 to 1; a rare event has a subjective probability close to 0, a very common event has a subjective probability close to 1.  A person's subjective probability of an event describes his/her degree of belief in the event.  Absent a probability model, only verbal statements can be used to describe a fingerprint examiner’s degree of belief for positive identification, i.e. bordering on certainty (with a reasonable degree of certainty), highly likely, likely, and so on.   

Subjective probability thresholds can be tested based on the following theory:

The more you are willing to pay for entering a bet in which you win some fixed amount if your belief turns out to be true, the higher is your subjective probability.

In the above theory, your willingness to bet serves as an indicator of how likely you think the belief is to be true.  For example, how much are you, a fingerprint examiner for the case at hand, willing to bet that the fingerprint identification is correct and not an unexpected look-alike or a clerical error?  Are you willing to bet 1 year in prison?  Are you willing to bet 20 years in prison?  Life in prison?  Are you willing to bet your life?

It is significant to note that the subjective approach to assigning probability value is used when prior and empirical approaches cannot be used.


Priori Probability (Classic or Theoretical)

Priori probability is a probability value that can be determined prior to any experimentation or trial. For example, the probability of obtaining a tail in tossing a coin once is fifty percent. The coin is not actually tossed to determine this probability.  No experiments are carried out.  It is simply observed that there are two faces to the coin, one of which is tails and that heads and tails are equally likely.  A priori probability is a situation where probability is assigned based on prior knowledge of the process involved.  Examples of this are that we can assign probability in card games, coin flipping, and die tossing.  This is sometimes called the classical method of assigning probability.

Empirical Probability (Frequentist)

Empirical probability is a relative frequency method, which determines probability value by observation and experimentation. An example of this is a manufacturing process where after checking one hundred parts, five are found defective. If the sample of one hundred parts was representative of the total population, then the probability of finding a defective part is .05 (5/100).  The question may be asked:  How is it known that this sample is representative of the total population?  If repeated trials average .05 defective, with little variation between trials, then it can be said that the empirical probability of a defective part is .05. 

The empirical approach to assigning probability is used when data is available about the past history of the experiment.  The probability of an outcome is the relative frequency of the outcome.  You can form this probability by taking the ratio of the number of times the outcome came up over the total number of times of the experiment.  As an example, if one takes a random sampling of 100 fingerprints and let's say it is found that a total of 100,000 ridge units, i.e. continuous ridge units, ending ridge units, bifurcating ridge units, and so on, are observed, and more specifically one observes 87,000 continuous ridge units, 8000 ending ridge units, 4000 bifurcating ridge units, and 1000 single ridge units (dots) are found, then the empirical probability of finding 1 ending ridge unit is 8000/100000, or 1 out of 12.5.  Similarly, the empirical probability of finding a bifurcating ridge unit is 1/25, and for a dot it is 1 out of 100.

Empirical probability can be thought of as the most accurate scientific "guess" based on the results of experiments to collect data about an event.  Because some problems are so complicated for analysis, we can only estimate probabilities from experience and observation. This is empirical probability.  Experience has shown that empirical probabilities, if carefully determined on the basis of adequate statistical samples, can be applied to large groups with the result that probability and relative frequency are approximately equal. By adequate samples we mean a large enough sample so that accidental runs of "luck," both good and bad, cancel each other. With enough trials, predicted results and actual results agree quite closely.

The empirical approach to determining probabilities relies on data from actual experiments to determine approximate probabilities instead of the assumption of equal likeliness. Probabilities in these experiments are defined as the ratio of the frequency of the occurrence of an event to the number of trials in the experiment.  If an experiment involves flipping a coin, the empirical probability of heads is the number of heads divided by the total number of flips.  The relationship between these empirical probabilities and the theoretical or true probabilities is suggested by the Law of Large Numbers.

Law of Large Numbers

The Law of Large Numbers states that as the number of trials of an experiment increases, the empirical probability approaches the theoretical probability [6][see Ridge Unit Frequency]. This makes sense as we would expect that if we roll a die numerous times, each number would come up approximately 1/6 of the time.

The study of empirical probabilities is known as statistics.  The T Model utilizes, in most part, the frequency method of empirical probability based on observation and experimentation to define relative probability values for ridge formation shapes in position.   It is significant to mention here that based on a sufficiently large fingerprint sampling, definitive probability values for all ridge formation types in position can be established.

"We use probability only in default of certainty."

Ludwig Wittgenstein [99]

Probability Theory and Fingerprint Identification 

Probability theory and the ability to reliable predict the future with numbers has applications for the field of fingerprint identification because it can reliably predict what is the chance for a look-alike to occur if all parts of all fingerprints in a population group were compared.

Based on empirical data gathered as a result of decades of operational experience, fingerprint look-alikes occur.  Small look-alikes, or arrangements of a few matching ridge details from different people, are relatively common whereas large look-alikes, or arrangement of large numbers of matching ridge details from different people, are unexpected and are not normally revealed without the use of an Automated Fingerprint Identification System (AFIS).  The power of AFIS reveals the unexpected look-alike that is otherwise unlikely to occur during the comparison of random fingerprint samples.  AFIS seeks out and finds the largest and best look-alikes in its database and thereby reveals the unexpected look-alike or outlier that is only made possible by it search capability. 

The Clark Non-Match and Chesapeake IAFIS look-alike are examples of two of the most unexpected fingerprint look-alikes, greatest fingerprint outliers or “black swans”, ever found as a result of an AFIS search.  The T Model successfully identified each of these largest and best look-alikes as insufficient to infer positive identification.  As a result it may be said that the T Model successfully calculates probabilities for both the expected and unexpected look-alikes in a given population.

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Probability theory and the ability to reliably predict the future with numbers is explained by Dr. Keith Devlin.

Probability theory is “the likelihood of the occurrence of any event in the doctrine of chances, or the ratio of the number of favorable chances to the whole number of chances, favorable and unfavorable”.