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A Priori Probability

A probability that is deduced from formal reasoning

What is A Priori Probability?

A priori probability, also known as classical probability, is a probability that is deduced from formal reasoning. In other words, a priori probability is derived from logically examining an event. A priori probability does not vary from person to person (as would a subjective probability) and is an objective probability.

 

A Priori Probability

 

Formula for A Priori Probability

 

A Priori Probability

 

Where:

  • f refers to the number of desirable outcomes.
  • N refers to the total number of outcomes.

 

Note that the formula above can only be used for events where outcomes all have equal odds of occurring and are mutually exclusive.

 

Example of Formal Reasoning in A Priori Probability

A priori probability requires formal reasoning. For example, consider a coin toss. What is the a priori probability of a head in a single coin toss?

One can argue that given a coin has two sides, both of which have equal surface areas, that it is symmetrical. Ignoring the possibility of a coin landing on its edge and staying there, it would suggest that the probability of a coin landing on heads is the same as a coin landing on tails. Therefore, the a priori probability of a coin toss landing on heads is equal to a coin toss landing on tails, which is 50%.

 

Examples of A Priori Probability

The following are examples of a priori probability:

 

Example 1: Fair Dice Roll

A six-sided fair dice is rolled. What is the a priori probability of rolling a 2, 4, or 6, in a dice roll?

The number of desired outcomes is 3 (rolling a 2, 4, or 6), and there are 6 outcomes in total. The a priori probability for this example is calculated as follows:

A priori probability = 3 / 6 = 50%. Therefore, the a priori probability of rolling a 2, 4, or 6 is 50%.

 

Example 2: Deck of Cards

In a standard deck of cards, what is the a priori probability of drawing an ace of spades?

The number of desired outcomes is 1 (an ace of spades), and there are 52 outcomes in total. The a priori probability for this example is calculated as follows:

A priori probability = 1 / 52 = 1.92%. Therefore, the a priori probability of drawing the ace of spades is 1.92%.

 

Example 3: Coin Toss

John is looking to determine the a priori probability of landing a head. He conducts a single coin toss, shown below:

Experiment 1

Result: Head

What is the a priori probability of landing a head?

The above is a trick example – the prior coin toss has no impact on the a priori probability of landing a head. The a priori probability of landing a head is calculated as follows:

A priori probability = 1 / 2 = 50%. Therefore, the a priori probability of landing a head is 50%.

 

Other Types of Probabilities

Apart from a priori probability, there are two other main types of probabilities:

 

1. Empirical Probability

Empirical probability refers to a probability that is based on historical data. For example, if three coin tosses yielded a head, the empirical probability of getting a head in a coin toss is 100%.

 

2. Subjective Probability

Subjective probability refers to a probability that is based on experience or personal judgment. For example, if the analyst believes that “there is an 80% probability that the S&P 500 will hit all-time highs in the next month,” he is using subjective probability.

 

Related Readings

CFI offers the Financial Modeling & Valuation Analyst (FMVA)™ certification program for those looking to take their careers to the next level. To keep learning and advancing your career, the following resources will be helpful:

  • Basic Statistics Concepts in Finance
  • Empirical Probability
  • Independent Events
  • Normal Distribution

Financial Analyst Certification

Become a certified Financial Modeling and Valuation Analyst (FMVA)® by completing CFI’s online financial modeling classes and training program!