A Priori Probability

What is A Priori Probability?

“A Priori Probability”, also known as classicial probability,  refers to the probability of those events that can only have a finite number of outcomes and each outcome is equally likely to occur. In this type of probability, the outcomes are not influenced by their preceding outcomes and any outcome drawn today will in no way influence the prediction of the probability of the future outcomes.

Explanation

The term “a priori” is Latin for the words “presumptive” or “deductive.” So, as the name suggests, it is more deductive and is not at all influenced by what has happened in the past. In other words, the underlying principle of a priori probability follows logic rather than history to determine the probability of a future event. Typically, the outcome of a classical probability is calculated by evaluating the pre-existing information or circumstance associated with a situation in a rational way. As already mentioned above, in such a probability estimation, each event is independent, and their previous events impact their occurrence in no way.

Formula

A-Priori-Probability-Formula

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The formula is expressed by dividing the number of desired outcomes by the total number of outcomes. Mathematically, it is represented as below,

A Priori Probability Formula = No. of Desired Outcomes / Total No. of Outcomes

It should be noted that the above formula can only be used in the case of events wherein all the outcomes equally likely to occur and are mutually exclusiveMutually ExclusiveMutually exclusive refers to those statistical events which cannot take place at the same time. Thus, these events are entirely independent of one another, i.e., one event's outcome has no impact on the other event's result.read more.

Examples

Below are examples to understand the concept in a better manner.

Example #1

Let us take the example of a fair dice roll to illustrate the concept. A fair dice has six sides with equal probability of rolling, and all the outcomes are mutually exclusive. Determine the a priori probability to roll a 1 or 5 in a fair dice roll.

Given,

  • No. of desired outcomes = 2 (roll a 1 or 5)
  • Total no. of outcomes = 6 (roll a 1, 2, 3, 4, 5 or 6)

Solution

Now, the probability of rolling a 1 or 5 in fair dice roll can be calculated by using the above formula as,

A Priori Probability Example 1
  • = 2 / 6
  • = 33.3%

Therefore, the probability of rolling a 1 or 5 in a fair dice roll is 33.3%.

Example #2

Let us take the example of a standard 52-card deck to illustrate the concept. There are 52 cards equally distributed among four suits (13 ranks in each suit) in a typical 52- card deck. If one draws one card and places it back in the deck, then determine it to draw a card from the hearts suit?

Given,

  • No. of desired outcomes = 13 (as each suite has 13 ranks)
  • Total no. of outcomes = 52

Solution

Now, the a priori probability of drawing a card from hearts suit can be calculated by using the above formula as,

Example 2
  • = 13 / 52
  • = 25.0%

Therefore, the probability of drawing a card from a heart suit from a standard deck is 25.0%.

Example #3

Let us take the example of a coin toss to illustrate the concept. A coin has two sides – a head and a tail. Determine the a priori probability to land a head in a usual coin toss.

Given,

  • No. of desired outcomes = 1 (land a head)
  • Total no. of outcomes = 2 (land a head or a tail)

Solution

Now, the probability of landing a head in a coin toss can be calculated by using the above formula as,

Example 3
  • = 1 / 2
  • = 50.0%

Prior Probability vs. A Priori Probability

Prior Probability vs A Priori Probability

Advantages

Some of the major advantages are as follows:

  • The concept of a priori probability is easy to explain.
  • It is a simple concept that can be applied to many real-life situations.

Drawbacks

Some of the major drawbacks are as follows –

  • It fails when the probability of occurrence of the events are not equally likely.
  • It can’t be used for cases where the number of outcomes is potentially infinite.

Conclusion

So, it can be seen that a priori probability is an essential statistical technique that also extends to other concepts. However, it has its own set of limitations that one needs to take cognizance of while drawing statistical insights.

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