Joint Probability Formula

What is the Joint Probability?

Joint Probability is the possibility of occurring one or more independent events at the same time, denoted as P (A∩B) or P (A and B) and is calculated by multiplying the probability of both the outcomes = P (A)*P (B)

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

Step 1- Find the Probability of Two events separately

Step 2 – To calculate joint probability, both the probabilities must be multiplied.

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 balls 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 below given data for calculation

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

Probability of choosing 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 students strength of 50 in a class, and 4 students are between 140-150cms in height. If you randomly select one student and without replacing the first selected person, you are selecting the second person what is a probability of both being between 140-150cms.

Solution

Use 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 the second person between 140-150cms without replacing the selected. As we already selected 1 from 4 the balance will be 3 students.

Probability of choosing 2 student

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 how they are choosing a course. There were two options, either by the quality of a college or by the cost, of course. Let’s find the joint probability if both full-timers and part-timers choose cost as the deciding factor.

Solution

Use below given data for 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

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 occurring one or more independent events at the same time. 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 has to occur, then the other event is already known, or true, then it is called a Conditional Probability. e.g., if event y has to be, then the 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. It can also be said as one event is dependent on the occurrence or existence of another event.

Both conditional and joint probabilities deal with two events, but their occurrence makes it different. In conditional, it has an underlying condition, whereas in joint, it just occurs at the same time.

Let’s consider an example if the price of crude oil increases, then there will be an increase in the price of petrol as well as in gold. If both gold and petrol prices increase at the same time, it can be said as joint probability, but with joint probability, we can’t measure how much one influences the other, there comes conditional probability it can be used to measure how much one event influence the other.

Relevance and Use

When two are more events occurring at the same time, the joint probability is used, mostly used by statisticians to indicate the likelihood of two or more events occurring same time, but it does not how they influence each other.

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

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