Conditional probability & distribution

< List of probability definitions


What is conditional probability?

Conditional probability is the probability of an event occurring, given that another event has already occurred. It is used to determine the probability of one event happening when we have information about the occurrence of a related event. For example, what is the probability the temperature will dip below freezing today, given that there is a cold front on the way?

Conditional probability notation and formula

Conditional probability is denoted as P(A|B), which is read as “the probability of event A happening given that event B has occurred.”

It is calculated by dividing the probability of the two events occurring together by the probability of the first event occurring. In mathematical terms, conditional probability can be calculated using the following formula:

P(A|B) = P(A ∩ B) / P(B)


  • P(A|B) represents the conditional probability of event A occurring, given that event B has occurred.
  • P(A ∩ B) denotes the probability of both events A and B happening together (also known as joint probability).
  • P(B) is the probability of event B occurring.

Note that conditional probability is only meaningful when event B has a non-zero probability, i.e., P(B) > 0.

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Conditional probability examples

Example 1: the Coin Toss

conditional probability example

Consider the probability of obtaining heads when flipping a coin, which is 0.5. The likelihood of getting heads twice consecutively is 0.25. If you already got heads on the first flip, the conditional probability of getting heads again on the second flip is calculated as 0.25 / 0.5 = 0.5.

This means that, given you obtained heads on the first flip, there is a 50% chance of getting heads on the subsequent flip.

Example 2: Car buyers

In a group of 100 car buyers, 40 purchased alarm systems, 30 opted for vegan leather seats, and 20 bought both an alarm system and vegan leather seats. If we randomly select a car buyer who acquired an alarm system, what is the likelihood they also purchased vegan leather seats?

  1. Determine P(A). This is given in the problem as 40%, or 0.4.
  2. Identify P(A∩B). This represents the intersection of A and B, where both events occur together. In this case, 20 out of 100 buyers, or 0.2.
  3. Plug your values into the formula: P(B|A) = P(A∩B) / P(A) = 0.2 / 0.4 = 0.5.

Thus, given that a buyer bought an alarm system, there is a 0.5 (50%) probability that they also purchased vegan leather seats.

More examples of conditional probability — from YouTube’s Prof. Essa.

Real life examples of conditional probability

Conditional probability is used in various real-life situations to make informed decisions and predictions, especially when events are interrelated. Here are a few examples:

  1. Medical Diagnosis: Doctors use conditional probability to determine the likelihood of a patient having a particular disease, given specific symptoms or test results. This helps them make more accurate diagnoses and treatment recommendations. For example, a woman aged between 40 and 50 years has roughly a 1% probability of breast cancer. However, if a woman receives a positive mammogram result, this probability alters: the chances of a woman having cancer with a positive mammogram result increase to roughly 8.3%. [1].
  2. Weather Forecasting: Meteorologists use conditional probability to predict the chance of rain or other weather events, given certain atmospheric conditions, such as temperature and humidity.
  3. Insurance: Insurance companies use conditional probability to calculate the risk of accidents or damage based on factors like age, driving history, and location. This information helps them set appropriate insurance premiums for their clients.
  4. Finance: In the stock market, investors use conditional probability to estimate the likelihood of stock price fluctuations given specific economic indicators, such as interest rates or unemployment rates.
  5. Marketing: Businesses use conditional probability to analyze customer behavior and preferences. For example, they might evaluate the probability of a customer purchasing a particular product, given that they have already purchased a related item. This information helps businesses develop targeted marketing strategies and product recommendations.
  6. Quality Control: In manufacturing, conditional probability can be used to determine the likelihood of a product defect, given specific production parameters, such as machine settings or raw material quality. This information helps companies improve their production processes and maintain high-quality standards.

These examples demonstrate how conditional probability plays a critical role in various industries, helping professionals make better decisions based on the relationships between different events.

Where does conditional probability come from?

The origins of conditional probability can be traced back to the 17th century when English mathematician Thomas Bayes devised a theorem that established the groundwork for probability theory. Bayes’ theorem asserts that the probability of an event happening, given that another event has already occurred, can be determined by dividing the joint probability of both events by the probability of the initial event.

In the 18th century, French mathematician Pierre-Simon Laplace formulated a method for computing conditional probabilities, which is still in use today. Laplace’s approach is centered around the concept of conditional expectation – the anticipated value of a random variable when another random variable has already been observed.

During the 19th century, German mathematician Carl Friedrich Gauss created a method for calculating conditional probabilities based on the least squares concept. Gauss’s technique remains widely used in various applications such as data analysis and machine learning.

In the 20th century, the advent of computers enabled the easier and faster computation of conditional probabilities. This development led to new applications for conditional probability in fields like weather forecasting, medical diagnosis, and risk assessment.

Currently, conditional probability is an integral element of probability theory and statistics, with applications spanning across weather forecasting, medical diagnosis, risk assessment, marketing, and game theory.

Key milestones in the history of conditional probability include:

  • 1763: Thomas Bayes publishes his theorem on conditional probability.
  • 1774: Pierre-Simon Laplace introduces a method for calculating conditional probabilities.
  • 1809: Carl Friedrich Gauss develops a method for calculating conditional probabilities based on least squares.
  • 20th century: The emergence of computers facilitates easier and quicker calculations of conditional probabilities.
  • Present day: Conditional probability is a fundamental concept in probability theory and statistics, employed in a wide array of applications.

The conditional probability formula we use today stems from the multiplication rule: P(A and B) = P(A) * P(B|A).

Here’s a step-by-step breakdown of how to derive the conditional probability equation from the multiplication rule:

  1. Write down the multiplication rule: P(A and B) = P(A) * P(B|A)
  2. Divide both sides of the equation by P(A): P(A and B) / P(A) = [P(A) * P(B|A)] / P(A)
  3. Cancel out P(A) on the right side of the equation: P(A and B) / P(A) = P(B|A)
  4. Rearrange the equation: P(B|A) = P(A and B) / P(A)

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Conditional probability distribution

A conditional probability distribution represents the probability distribution of a random variable, determined based on the principles of conditional probability after observing the outcome of another related random variable.

Put simply, it refers to the likelihood of a specific event happening, provided that another event has already taken place. It can also be thought of as a probability distribution for a specific subgroup. Essentially, it outlines the likelihood of a randomly chosen individual from that subgroup possessing a particular attribute of interest.

For instance, imagine a biased coin with a 60% chance of landing on heads. Flipping the coin once yields a 0.6 probability of it landing on heads. If the coin is flipped twice and lands on heads both times, the probability of it landing on heads the third time remains 0.6 since the first two flips do not influence the third flip’s probability.

Now, consider a deck of cards where the probability of drawing a certain card is 1/52. If we draw a card and it is not the desired one, the probability of drawing the sought-after card in the next draw is still 1/52, as the first draw does not affect the second draw’s probability.

However, if we know that the first draw resulted in a red card, the probability of the second draw yielding a black card rises to 26/51 because there are now 26 black cards and 51 cards in total left in the deck.

In this case, the probability of drawing a black card on the second draw is conditional upon the fact that a red card was drawn on the first draw, forming a conditional probability distribution.

Conditional probability distribution definition

More formally, a conditional probability distribution is defined as follows [3]: Let and Y be discrete random variables with joint probability mass function (PMF) given by p(x, y). Then the conditional PMF of X, given that Y = y, is denoted by pX|Y(x | y) and given as

conditional probability distribution

Conditional probability distribution applications

Conditional probability distributions have various applications, such as:

  • Machine learning: They are used in machine learning algorithms for predicting future events.
  • Finance: They help in determining the risk of investments.
  • Healthcare: They aid in estimating the likelihood of a patient developing a specific disease.

Overall, conditional probability distributions serve as a useful tool for predicting future events and assessing the risk of particular outcomes.

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The Array Distribution

The array distribution refers to the conditional distribution of independent random variables X1 (given X2, … , Xn). It is often used as a synonym for a conditional distribution (e.g., Kotz et al. [4], Giri & Banerjee [5], Wadworth & Bryan [6]). You’re more likely to see the term “array distribution” in applied statistics; in theoretical statistics, the usual term is conditional distribution in theoretical statistics [7].

The Encyclopedia of Statistical Sciences [8] gives an example of an array distribution of X:

The variance of this distribution is called the array variance; if the variance doesn’t depend on X2, … ,Xn , the variation is homoscedastic [9]. Its variance-covariance matrix is called the array-variance-covariance matrix.

The array distribution is denoted by:

You can think of this as the density of y for a fixed value of x.


[1] ICMA Photos, CC BY-SA 2.0, via Wikimedia Commons

[2] Bayes’ Formul

[3] LibreTexts Statistics. 5.3: Conditional Probability Distributions. Retrieved June 10, 2023 from:’s_College_Notre_Dame/MATH_345__-Probability(Kuter)/5%3A_Probability_Distributions_for_Combinations_of_Random_Variables/5.3%3A_Conditional_Probability_Distributions

[4] Kotz, S. et al. (2000). Continuous Multivariate Distributions, Volume 1 Models and Applications. John Wiley & Sons.

[5] Giri, P. & Banerjee, J. (2021). Statistical Tools and Techniques. Academic Publishers.

[6] Wadsworth, G. & Bryan, J. (1960). Introduction to Probability and Random Variables. McGraw-Hill.

[7] Vidakovic , B. et al. (Eds.) (2005). Encyclopedia of Statistical Sciences, Volume 3. Wiley.

[8] Haight (ed.) (1963). Mathematical Theories of Traffic Flow. Elsevier Science

[9] Johnson, Kotz, and Balakrishnan, (1994), Continuous Univariate Distributions, Volumes I and II, 2nd. Ed., John Wiley and Sons.

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