A Priori Probability & Example

What Is a Priori Probability?

A priori probability refers to the likelihood of an event occurring when there is a finite amount of outcomes and each is equally likely to occur. The outcomes in a priori probability are not influenced by the prior outcome. Or, put another way, any results to date will not give you an edge in predicting future results. A coin toss is commonly used to explain a priori probability. The probability of ending with heads or tails is 50% with each coin toss regardless of whether you have a run of heads or tails. The largest drawback to this method of defining probabilities is that it can only be applied to a finite set of events as most real-world events we care about are subject to conditional probability to at least some degree. A priori probability is also referred to as classical probability.

Understanding A Priori Probability

A priori probability is largely a theoretical framework for probabilities that can be constrained to a small number of outcomes. The formula for calculating a priori probability is very straightforward:

A Priori Probability = Desired Outcome(s)/The Total Number of Outcomes

So the a priori probability of rolling a six on a six-sided die is one (the desired outcome of six) divided by six. So you have a 16% chance of rolling a six and the exact same chance with any other number you choose on the dice. A priori probabilities can be stacked within the outcome set, of course, so your odds of rolling an even number on the same die increases to 50% simply because there are more desired outcomes.

Real World Example of A Priori Probability

An everyday example of a priori probability is your chances of winning a numbers-based lottery. The formula for calculating the probability becomes much more complex as your chances are based on the combination of numbers on the ticket being randomly selected in the correct order, and you can buy multiple tickets with multiple number combinations. That said, there are an finite selection of combinations that will result in a win. Unfortunately, the number of possible outcomes dwarfs the number of desired outcomes — your particular set of tickets. The probability of winning the grand prize in a lottery like the Powerball Lottery in the U.S. are one in hundreds of millions. Moreover, the chances of winning the grand prize exclusively (not splitting) go down as the pot goes up and more people play.

A Priori Probability and Finance

The application of a priori probability to finance is limited. Outside of discouraging people from putting their financial fate in the hands of the lottery, most outcomes that people in finance care about do not have a finite number of outcomes. You cannot say that a stock's price has three possible outcomes of going up, down, or staying flat when these outcomes are influenced by a range of outside factors that change the likelihood of each outcome.

In finance, people more commonly use empirical or subjective probability as opposed to classical probability. In empirical probability, you look at past data to get an idea of what future outcomes will be. In subjective probability, you overlay your own personal experiences and perspectives over the data to make a call that is unique to you. If a stock has been on a tear for three days after outperforming analysts' recommendations, an investor may reasonably expect it to continue based on the recent price action. However, another investor may see the same price action and remember that consolidation followed a steep rise in this stock two years ago, taking the opposite message from the same price data. Depending on the market, both investors could be no more accurate than a prediction via a priori probability, but we feel better about decisions we can justify with at least some logic beyond random chance.