## 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.

### Formula for 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 CFI resources will be helpful: