Empirical probability is based on actual occurrences of events, while estimated probability is based on general principles. Empirical probability becomes more reliable with more data, and can be used in economics and finance to inform decisions. Theoretical and empirical probabilities can diverge, indicating potential issues with the model used for theoretical probability.
Empirical probability is a probability calculation based on the actual occurrence of a certain type of event. It is different from estimated, or theoretical, probability, which produces a value based on general principles rather than observed facts. Empirical probability describes a more inductive process, one that decreases the error resulting from incorrect models but increases the error resulting from random events.
A simple example to understand the two types of probabilities is a simple repeated coin toss. Let’s say a coin is tossed 100 times. It comes up heads 54 times and tails 46 times. There are two different ways to estimate the probability that the next toss will come up heads. The theoretical probability is 50 percent. This probability remains constant from one pull to another. The empirical probability, on the other hand, is 54%. The coin has come up heads 54% of the time so far; based on this data alone, one might expect it to be slightly more likely to emerge again. The empirical probability changes with the arrival of new data. If after 200 tosses the coin has come up heads 104 times, the empirical probability that the next coin will be heads is now 52%.
Empirical probabilities become more reliable the more data there is. If the model for producing the theoretical probability is good, in the example above, if the coin is fair, the theoretical and empirical probabilities will converge as the sample size increases. After a million coin tosses, an observer should expect the empirical probability to be very close to the predicted probability, 50%.
The more the two types of probability diverge, the more an observer can consider changing the parameters of his model for theoretical probability. In the classical gambler’s fallacy, where a coin comes up heads 99 times, a basic math textbook will say that the next coin still has a 50% chance of being tails. This answer is based on the assumption that the coin is fair: that it has an even distribution of weight and air resistance, that it is tossed effectively and randomly, and so on. The estimated probability could tell the player in this situation that the coin is not fair. The extreme deviation from the theoretical probability suggests that there may be something wrong with one of the assumptions used to calculate it.
The empirical probability does not always have to be twice the theoretical probability. It could be used to calculate the probability of an event about which little else is known. For example, if a person flips over a two-sided object whose two sides have different properties, he might rely more on an empirical element of the probability that it will land on a certain side. Again, the more data you have, the higher the quality of your empirical calculation.
People in the fields of economics and finance could use empirical probability to help inform their decisions. An economist, after creating a theoretical model of a market, should want to compare her calculations with an empirical calculation of the probabilities involved. She might rely heavily on empirical probabilities to fill in the coefficients in her model that she might have no other way of calculating. In practice, useful economic models almost always combine elements of theoretical and empirical probability.
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