Conditional Probability: Formula and Real-Life Examples

Conditional Probability Definition

Investopedia / Madelyn Goodnight

What Is Conditional Probability?

Conditional probability is defined as the likelihood of an event or outcome occurring, based on the occurrence of a previous event or outcome. Conditional probability is calculated by multiplying the probability of the preceding event by the updated probability of the succeeding, or conditional, event.

Conditional probability can be contrasted with unconditional probability. Unconditional probability refers to the likelihood that an event will take place irrespective of whether any other events have taken place or any other conditions are present.

Key Takeaways

  • Conditional probability refers to the chances that some outcome occurs given that another event has also occurred.
  • It is often stated as the probability of B given A and is written as P(B|A), where the probability of B depends on that of A happening.
  • Conditional probability can be contrasted with unconditional probability.
  • Probabilities are classified as either conditional, marginal, or joint.
  • Bayes' theorem is a mathematical formula used in calculating conditional probability.

Understanding Conditional Probability

Conditional probabilities are contingent on a previous result or event occurring. A conditional probability would look at such events in relationship with one another. Conditional probability is thus the likelihood of an event or outcome occurring based on the occurrence of some other event or prior outcome.

Two events are said to be independent if one event occurring does not affect the probability that the other event will occur. However, if one event occurring or not does, in fact, affect the probability that the other event will occur, the two events are said to be dependent. If events are independent, then the probability of some event B is not contingent on what happens with event A. A conditional probability, therefore, relates to those events that are dependent on one another.

Conditional probability is often portrayed as the "probability of A given B," notated as P(A|B).

Conditional probability is used in a variety of fields, such as insurance, economics, politics, and many different fields of mathematics.

Conditional Probability Formula

P(B|A) = P(A and B) / P(A)

Or:

P(B|A) = P(A∩B) / P(A)
Where
P = Probability
A = Event A
B = Event B

Unconditional probability is also known as marginal probability and measures the chance of an occurrence ignoring any knowledge gained from previous or external events. Since this probability ignores new information, it remains constant.

Examples of Conditional Probability

As an example, suppose you are drawing three marbles—red, blue, and green—from a bag. Each marble has an equal chance of being drawn. What is the conditional probability of drawing the red marble after already drawing the blue one?

First, the probability of drawing a blue marble is about 33% because it is one possible outcome out of three. Assuming this first event occurs, there will be two marbles remaining, with each having a 50% chance of being drawn. So the chance of drawing a blue marble after already drawing a red marble would be about 16.5% (33% x 50%).

As another example to provide further insight into this concept, consider that a fair die has been rolled and you are asked to give the probability that it was a five. There are six equally likely outcomes, so your answer is 1/6.

But imagine if before you answer, you get extra information that the number rolled was odd. Since there are only three odd numbers that are possible, one of which is five, you would certainly revise your estimate for the likelihood that a five was rolled from 1/6 to 1/3.

This revised probability that an event A has occurred, considering the additional information that another event B has definitely occurred on this trial of the experiment, is called the conditional probability of A given B and is denoted by P(A|B).

Another Example of Conditional Probability

As another example, suppose a student is applying for admission to a university and hopes to receive an academic scholarship. The school to which they are applying accepts 100 of every 1,000 applicants (10%) and awards academic scholarships to 10 of every 500 students who are accepted (2%).

Of the scholarship recipients, 50% of them also receive university stipends for books, meals, and housing. For the students, the chance of them being accepted and then receiving a scholarship is .2% (.1 x .02). The chance of them being accepted, receiving the scholarship, then also receiving a stipend for books, etc. is .1% (.1 x .02 x .5).

Conditional Probability vs. Joint Probability and Marginal Probability

  • Conditional probability: p(A|B) is the probability of event A occurring, given that event B occurs. For example, given that you drew a red card, what’s the probability that it’s a four (p(four|red))=2/26=1/13. So out of the 26 red cards (given a red card), there are two fours so 2/26=1/13.
  • Marginal probability: the probability of an event occurring (p(A)) in isolation. It may be thought of as an unconditional probability. It is not conditioned on another event. Example: the probability that a card drawn is red (p(red) = 0.5). Another example: the probability that a card drawn is a 4 (p(four)=1/13).
  • Joint probability: p(A ∩B). Joint probability is that of event A and event B occurring. It is the probability of the intersection of two or more events. The probability of the intersection of A and B may be written p(A ∩ B). Example: the probability that a card is a four and red =p(four and red) = 2/52=1/26. (There are two red fours in a deck of 52, the 4 of hearts and the 4 of diamonds).

Bayes' Theorem and Conditional Probability

Bayes' theorem, named after 18th-century British mathematician Thomas Bayes, is a mathematical formula for determining conditional probability. The theorem provides a way to revise existing predictions or theories (update probabilities) given new or additional evidence. In finance, Bayes' theorem can be used to rate the risk of lending money to potential borrowers.

Bayes' theorem is also called Bayes' Rule or Bayes' Law and is the foundation of the field of Bayesian statistics. This set of rules of probability allows one to update their predictions of events occurring based on new information that has been received, making for better and more dynamic estimates.

Bayes' theorem is well suited to and widely used in machine learning.

How Do You Calculate Conditional Probability?

Conditional probability is calculated by multiplying the probability of the preceding event by the probability of the succeeding or conditional event. Conditional probability looks at the probability of one event happening based on the probability of a preceding event happening.

What Is a Conditional Probability Calculator?

A conditional probability calculator is an online tool that will calculate conditional probability. It will provide the probability of the first event and the second event occurring. A conditional probability calculator saves the user from doing the mathematics manually.

What Is the Difference Between Probability and Conditional Probability?

Probability looks at the likelihood of one event occurring. Conditional probability looks at two events occurring in relation to one another. It looks at the probability of a second event occurring based on the probability of the first event occurring.

What Is Prior Probability?

Prior probability is the probability of an event occurring before any data has been gathered to determine the probability. It is the probability as determined by a prior belief. Prior probability is a component of Bayesian statistical inference.

What Is Compound Probability?

Compound probability looks to determine the likelihood of two independent events occurring. Compound probability multiplies the probability of the first event by the probability of the second event. The most common example is that of a coin flipped twice and the determination if the second result will be the same or different than the first.

The Bottom Line

Conditional probability examines the likelihood of an event occurring based on the likelihood of a preceding event occurring. The second event is dependent on the first event. It is calculated by multiplying the probability of the first event by the probability of the second event.

Article Sources
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  1. Harvard Catalyst. "Bayes Theorem."

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