Joint Probability Formula

Updated on March 13, 2024
Article byAiswarya Swaminathan
Edited byAshish Kumar Srivastav
Reviewed byDheeraj Vaidya, CFA, FRM

What is the Joint Probability?

A joint probability is a possibility of occurring one or more independent events simultaneously, denoted as P (A∩B) or P (A and B). One can calculate it by multiplying the probability of both outcomes = P (A)*P (B).

Joint Probability Formula = P (A∩B) = P (A)*P (B)

Key Takeaways

  • A joint probability is the probability of one or more independent events occurring simultaneously and is represented as P(A∩B) or P(A and B).
  • It can be calculated by multiplying the individual probabilities of the events: P(A) * P(B).
  • A joint probability is used to determine the likelihood of multiple events coinciding, but it does not provide information about causation or influence between the events.
  • Statisticians utilize joint probability to understand the probability of two or more events occurring together, considering their independent probabilities. However, it does not capture the relationship or influence between these events.
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Step 1- Find the probability of two events separately.

Step 2 – Both probabilities must be multiplied to calculate joint probability.

Joint-Probability-Formula

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Source: Joint Probability Formula (wallstreetmojo.com)

Examples of Joint Probability Formula (with Excel Template)

You can download this Joint Probability Formula Excel Template here – Joint Probability Formula Excel Template

Example #1

Let’s consider a simple example. A bag contains 10 blue and 10 red balls if we choose 1 red and 1 blue from the bag on a single take. What will be the joint probability of choosing 1 blue and 1 red?

Solution:

  • Possible outcomes = (red, blue),(blue, red),(red, red), (blue, blue)=4
  • Favorable outcomes = (red, blue) or (blue, red) = 1

Use the below-given data for the calculation.

ParticularsValue
Number of Red Balls10
Number of Blue Balls10
Probability of Choosing 1 Red and 1 Blue1
Possible Outcomes4

Probability of choosing the red ball

Joint Probability Formula Example 1.1
  • P (a) = 1/4
  • = 0.25

Probability of choosing a blue ball

Joint Probability Formula Example 1.2
  • P (b) = 1/4
  • = 0.25
Joint Probability Formula Example 1.3
  • =0.25*0.25
Joint Probability Formula Example 1.4

Example #2

You have a student’s strength of 50 in a class, and 4 students are between 140-150cms in height. If you randomly select one student without replacing the first selected person, you select the second person. What is the probability of both being between 140-150cms?

Solution

Use the below-given data for calculation.

ParticularsValue
Total no of students in class50
No of students between 140-150 cms height4

First, need to find the probability of choosing 1 student in the first draw

Joint Probability Formula Example 2.1
  • P(a) =50*4
  • =0.08

Next, we need to find a second person between 140-150cms without replacing the selected one. As we already selected 1 from 4 the balance will be 3 students.

Probability of choosing 2 students

Example 2.2
  • P(b) =50*4
  • =0.08
Joint Probability Formula Example 2.3
  • =0.08*0.0612

Therefore, the joint probability of both students being 140-150cms will be:

Example 2.4

Example #3

There was a survey with full-timers and part-timers in a college to find out how they choose a course. There were two options, either by the quality of a college or by the cost, of course. So let’s find the joint probability of both full-timers and part-timers choosing cost as the deciding factor.

Solution

Use the below-given data for the calculation.

ParticularsCostQualityTotal
Fulltime3070100
Part-time6050110
Total90120210

Probability of full-timers in college:

Example 3.1
  • =30/210
  • Full-timers = 0.143

Probability of part-timers in college:

Joint Probability Formula Example 3.2
  • =60/210
  • Part-timers = 0.286

The joint probability of full-timers and part-timers is calculated as follows:

Example 3.3
  • =0.143*0.286
Joint Probability Formula Example 3.4

Difference Between Joint, Marginal, and Conditional Probability

  • JOINT PROBABILITY – It is the possibility of simultaneously occurring one or more independent events. For instance, if an event Y appears and the same time event X appears, it is called a joint probability.
  • CONDITIONAL PROBABILITY –  If one event occurs, then the other event is already known or true, called conditional probability. e.g., if event y has to be, then event X must be true.

Conditional probability occurs when there is a conditional that the event already exists or the event already given has to be true. One can also say that one event depends on another event’s occurrence or existence.

  • MARGINAL PROBABILITY – It simply refers to the probability of occurrence of a single event. It does not depend on another probability of occurring, like conditional probability.

Both conditional and joint probabilities deal with two events, but their occurrence differs. In conditional, it has an underlying condition, whereas in joint, it occurs simultaneously.

For example, if the price of crude oil increases, then there will be an increase in the price of petrol and gold. So if gold and petrol prices increase simultaneously, it can be said to be a joint probability. Still, we can’t measure how much one influences the other with joint probability. So instead, one can use conditional probability to measure how much one event influences the other.

Relevance and Use

When two more events occur simultaneously, the joint probability is used. Mostly used by statisticians to indicate the likelihood of two or more events occurring simultaneously, but it does not know how they influence each other.

We can use it to know the value of both events, but it will not show how far one event will influence the other.

Frequently Asked Questions (FAQs)

1. What are the assumptions used in the joint probability formula?

The joint probability formula assumes that the events being considered are independent of each other. This means that the occurrence or outcome of one event does not affect the occurrence or outcome of the other event.

2. What are the limitations of the joint probability formula?

The joint probability formula assumes independence between events, which may not always hold true in real-world situations. If events are dependent on or influenced by each other, the joint probability formula may not accurately reflect their actual probabilities.

3. What is the significance of joint probability in decision-making and forecasting?

Joint probability plays a crucial role in decision-making and forecasting by providing insights into the likelihood of multiple events occurring together. It allows decision-makers to assess the combined probabilities of different outcomes and make informed choices based on the co-occurrence of events. In forecasting, joint probability helps estimate the likelihood of multiple variables aligning in a particular way, enabling more accurate predictions and planning.

This article has been a guide to Joint Probability and its definition. Here, we discuss the formula for calculating joint probability, practical examples, and a downloadable Excel template. You can learn more from the following articles: –