# Empirical Probability

## What Is Empirical Probability?

An empirical probablility, also called an experimental probability, is closely related to the relative frequency of an event. Empirical probability uses the number of occurrences of a given outcome within a sample set as a basis for determining the probability of that outcome occurring again. The number of times "event X" happens out of 100 trials will be the probability of event X happening.

### Key Takeaways

• The capital asset pricing model is the basis for most empirical probability studies using real market data.
• Empirical probability is based on a ratio of the number of attempts of a task to the number of a specific result (e.g., coin tosses to number of heads or tails achieved).
• Theoretical probability starts with the desired outcome (heads) and relates it to the number of possible results (heads or tails).
• Conditional probability looks at the likelihood of an event occurring based on the prior occurrence of another event (e.g., if I walk on the ice, what is the probability I will fall).
• The availability of large amounts of computation power in today's computers has made calculating probability easier and more common.

## Understanding Empirical Probability

For a theory to be proven or disproven, the researcher must collect empirical evidence. An empirical study is performed using actual market data. For example, many empirical studies have been conducted on the capital asset pricing model (CAPM), and the results are slightly mixed.

In some analyses, the CAPM model does hold in real-world situations, but most studies have disproven the model for projecting returns. For instance, the CAPM is often used to estimate a company's weighted average cost of capital. Although the model is not completely valid, that is not to say that there is no utility associated with using the CAPM.

### Empirical Probability Formula

The empirical probability formula creates a ratio of the number, times the desired event occurred, to the total number of times one tried to reach it. An example would be I rolled the dice three times and got 12 three times, for a statistical probability of 12/12 or 100%. This calculation demonstrates the flaw of empirical probability.

## Examples of Empirical Probability

Consider, for example, that you want to look at a small dataset such as the possibility of rolling a six when you roll a single die. If on the first roll you roll a 2, on the second a 5, and on the third a 4, the empirical probably is 0/3=0%. The empirical probability in this case is 0%.

If, for another example, you toss a coin three times looking for heads and get heads three times, the empirical probability of getting heads is 100% or 3/3=1000%.

Note that both of these examples, largely because of their sample size, will lead you to the wrong conclusion in both cases. Clearly, the probability of either side of a coin toss coming up is 1/2, while the die, having six sides, is 1/6.

## Empirical Probability vs. Theoretical Probability

Empirical probability is based on the ratio of the number of times an event occurs to the number of attempts made. It is based solely on that data, and thus can frequently produce inaccurate results, particularly where a small data set is used. Theoretical or classical probability defines a desired outcome and then creates a ratio of the number of successful outcomes to the total of the possible outcomes. Thus, a coin tossed once where T is for Tails would be P(E)=1/2.

## Other Types of Probability

Empirical probability is obviously not the only type of probability which can be calculated. There are several other types, each of which may be most useful in any given situation.

### Conditional Probability

Conditional probability is the likelihood that an event will occur based on the occurrence of some previous event or outcome. It is calculated by multiplying the probability (P) of the preceding event (PE) by the updated probability of the succeeding or conditional event (CE). It is shown as P=PE(PC).

### Subjective Probability

Subjective probability is anyone's best judgment or opinion as to the probability of a given event. Obviously, this is not ideal or even very scientific, but if there is no prior experience and no particular theory, it is sometimes the best option available.

### Axiomatic Probability

Axiomatic probability is a unifying theory of probability. It sets out a series of rules that apply to all types of probability calculations, based on Kolmogorov's Three Axioms. It is defined by three ideas:

Probability is a set function P(E) that says for every event E there is a number referred to as the "probability of E" such that: 1. The probability of an event is greater than or equal to zero: P(E)>0. 2. The probability of the same space is one P(Omega)=1.

### Classical or Theoretical Probability

Calculated without experimentation, classical or theoretical probability assumes that all outcomes of a given event are equally likely. It is calculated by defining an event then determining the probability of that event as a ratio of the number of successful outcomes to the total number of possible outcomes. Thus, if we toss a coin once and get the side S we wanted the formula would read P(S) = 1/2.

### Joint Probability

Joint probability measures the likelihood of two events occurring together and at the same point in time. In other words, joint probability is the probability of event 1 happening at the same time that event B happens. Since it is looking for the simultaneous occurrence of two events, there must be two observers. Joint probability is simultaneous; conditional probability is linear, meaning B will happen if A has already happened.

## The Bottom Line

Probability makes predictions in various ways to meet various needs. Given the vast increase of computing power, probability calculations of immense size are now possible and have changed the popularity and usefulness of differing kinds of probability.

## How Do You Calculate Empirical Probability?

You can calculate empirical probability by creating a ratio between the number of ways an event happened to the number of opportunities for it to have happened. In other words, 75 heads out of 100 coin tosses come to 75/100= 3/4. Or P(A)-n(a)/n where n(A) is the number of times A happened and n is the number of attempts.

## What Is the Difference Between Empirical Probability and Classical Probability?

The primary difference is that an empirical probability requires that probability experiment . One has to toss the coin X times to find out how many times heads or tails will come up. Classical probability is used without an experiment or where it isn't possible to perform an experiment and therefore all results may be equally likely.

## What Is Subjective Probability?

Subjective probability is essentially just what it says it is, someone's opinion of the probability that an event will occur. It may not seem like much, but if there is no experience and no theory, it may be the best available option.

## Is a Normal Distribution Theoretical or Empirical?

The Standard Normal Curve is theoretical distribution rather than an empirical distribution because it exists in theory rather than on an empirical experiment. It does not exactly correspond to any distribution occurring in the world.