# Strategy for Information Markets/Background/Probability in Information Markets

Probability is a concept that is extremely prevalent in the study of Information Markets and Economics in general. It is particularly useful when examining "Network Externalities" as well as Economic Prediction Models. The following will provide a brief background on probability and address its specific uses with regard to Information Markets.

## General

According to the Business Dictionary, probability is defined as the quantitative measure of the likelihood of something occurring. In other words probability is the chance that something happens in a certain way. This chance can be non-influenced or random, which is known as physical probability, or it can be influenced by certain factors which is known as evidential probability. Physical probability is somewhat of the classical idea of probability while evidential is more of a real world application. Physical probability will be given a brief overview in order to understand the general concepts however the large focus of this page will be toward evidential probability as this is the more relevant type of probability to information markets.

## Mathematical Definition and Basic Laws
### Main Laws

The concept or probability has two main laws or axioms that define it. They are:
* The probability of any occurrence is a real number between the value of 0 and 1 inclusive.
* The probability of an occurrence that never happens is 0 and the probability of an occurrence that always happens is 1.

### Notation

The probability of an occurrence is written as P(A) = X. This is read as the probability of event A occurring is equal to some value X. For example, if we were to measure the chance that the earth is round, we could say that given A=Earth is round,P(A) = 1 , since it is always true that the Earth is round. The

### Basic Situations

* Not Event: Since we know that the probability of an event A occurring is written as P(A), and that all probabilities must be between 0 and 1 inclusive, then we can say that the probability of even A not occurring is 1 - P(A).
* Multiple Events separate: If two events A and B are completely separate from one another, then the probability of both events occurring is written as P(A and B). This is equivalent to P(A)P(B). An example of separate multiple events would be if there were two pitchers in two different baseball games, and one wanted to know what the probability be that they both threw fastballs on the first pitch.

#### Mutually Exclusive

Mutually exclusive is a concept needed to understand the other basic situations. The term mutually exclusive signifies when two events are a possible result of a single occurrence. For example, in baseball when one is up to bat, the pitcher may throw a curveball or a fastball. The probability of getting a curveball or fastball are two events on the single occurrence of the first pitch at bat. This would be signified by the notation P(A or B) or P(A U B). This is equivalent to P(A) + P(B).
If two events of a single occurrence are not mutually exclusive, that implies that their is some overlap is the possible events. For example, if one was collecting baseball cards he or she wanted to know the probability of getting a rookie card or a pitcher's card. Since there will be cards for rookie pitchers these to events are not mutually exclusive. This situation is denoted by P(A) + P(B) - P(A and B). Conceptually this expression makes sense as you have to remove the probability of both happening so you don't count it twice.

## Physical Probability

Physical probability, also known as Classical probability is study of physical objects which have which have equally possible events for every occurrence. Examples of this include heads or tails on a coin, number sides on a die, or drawing a certain card in a card deck. It is probability at its basis. The classical theory of probability, written by Laplace is what governs this type of probability. It states that the probability of a favorable event occurring, is equal to the total number of favorable events, divided by the total number of possible events.. If one were to examine this using a simple coin toss, then the probability of the coin landing heads would be 1 possible favorable outcome, divided by 2 total possible outcomes (heads and tails). so using probability notation P(H) = 1/2 where H = a coin landing on heads when tossed.

## Evidential Probability

Evidential Probability has to do with the study of uncertain situations given certain knowledge or known factors. In other words, given a situation where the result is not known, one is provided with knowledge or evidence of why certain events may be the ending result, and from that evidence the probability of each event occurring is derived. Examples of this are the chances a man on trial will be convicted or acquitted. The probability of each of these outcomes is influenced by the evidence presented on trial and how each juror interprets that evidence. In the study of Information Markets, evidential probability is the type of probability that is most prevalent.

### Conditional Probability

To express a probability where one event is dependent on the given evidence of another, the concept of conditional probability is used. If one had two events A and B, a conditional probability would be the chance of event B occurring given that event A has occurred. This is expressed by the notation P(B|A), which is equivalent to P(A and B)/P(A).

## Common Applications of Probability in Information Markets
### Information Cascades

Information cascades are a very common application of probability in information markets. Cascades are centered around evidential probability and more specifically conditional probability. The goal of the cascade is to guess the correct decision based on the evidence of prior events. For a detailed analysis of information cascades, visit the discussion section in Network Externalities Two Bandwagons

### Prediction Markets

Prediction Markets are another area of study in Information markets where probability is applied. In prediction markets, prices are determined by the probability of a certain event occurring and the belief by individuals that a certain event will occur. For more information on Prediction Markets visit Prediction Markets.

## References

* ↑ business.yourdictionary.com, 2012
* 1 2 Strevens, Michael, Probability and Chance, The Encyclopedia of Philosophy, 2006
* 1 2 3 4 Omegamath.com 1998
* ↑ Laplace, Classical, Stats.org/uk , 1999
* ↑ Kyburg H., Evidential Probability, University of Rochester, 1991
* ↑ Easton V. and McColl's J., Conditional Probability, Stat.Yale.edu , 1997
* ↑ Wolfers, J. and Zitzewitz E., Prediction Markets, Journal of Economic Perspectives University of Pennsylvania, 2004