# The Difference Between Empirical and Theoretical Probability ••• vencavolrab/iStock/GettyImages

Working out the probability of something occurring is a mathematical problem that is frequently applied in the wider world, so understanding how it works could set you in good stead for the future. Estimates are used in business, science and finance to help people project what may happen in the coming months and years. That is what probability is all about -- making an educated guess as to what might happen in the future. There are different ways to estimate the probability of a particular occurrence coming to transpire and two of these are known as theoretical and empirical probability.

## Theoretical Probability

Theoretical probability, also known as a priori probability, is calculated before any event has taken place. For instance, if you were to roll a pair of dice, you could work out the theoretical probability of rolling a four before any dice had been rolled at all. Mathematicians do this through a simple equation. The number of possible outcomes is divided by the number of ways in which a particular outcome could be arrived at. There are 36 different possible results after throwing the dice; however, there are only three ways you could roll a four. The dice could land on one and three, two and two, or three and one. Thus, the probability of rolling a four when using two dice is 3/11.

## Empirical Probability

Empirical probability is calculated after the event has occurred. By observing the pattern of events and how often a certain outcome has been seen, mathematicians try to estimate how often they can expect to see a certain outcome in the future. If you tossed a coin twice and the first time it came up tails and the second time came up heads, you could assume that the probability the coin would land on head is 1/2. This is a very basic form of empirical probability, however, and has a high risk of being incorrect because a series of only two events (coin tosses) have been observed. Were you to toss the coin 100 times, you would get a clearer view of how probable it is that the coin lands on heads each time. The more data that can be analyzed, the more accurate your estimate is likely to be.

## Subjective Probability

Subjective probability is more connected to the original meaning of the word probable -- as similar to plausible -- than its mathematical application. This type of probability refers to a personal intuition or judgment as to what could happen, or what is probably true. It is used when other calculations of probability are uncertain and tends to be given by a person experienced in the field. For instance, a doctor may give an approximation of life expectancy.

## Practical Applications

The various types of probability have very different practical applications; in some cases, theoretical probability would give you a less accurate result than empirical probability and vice versa. Bookmakers are more likely to use empirical probability to give the odds on a horse, for example, because simply calculating the probability of any horse winning would be inaccurate given the differing performances of both animals and jockeys. Bookmakers are therefore more likely to look at past performance to decide the probability of a horse winning. If you were gambling with dice, however, you would be better off calculating the theoretical probability of the dice landing on a certain number, as each number of each die has an equal chance of turning up. Looking back on the past performance of the dice may be redundant.

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