What is joint probability?

The most fundamental uncertainties of the universe can be explained with the help of Mathematics. This universal language can quantify tangible things and lend meaning to the most complex puzzles. Does that mean to understand maths is to understand everything about the universe? Well, no. But it is a good starting point. One that may feel difficult to grasp initially but gets easier if the basics are well understood.

Several sub-topics within maths make the subject so interesting. One of which is probability.

What is Probability?

When two IPL captains decide who will bat or bowl first, the probability of making the right call is 50%. How? Because a coin will either land as Heads or Tails. 2 outcomes in one actionso, 50%. What about a dice while playing ludo? What’s the chance that a player can get six if only one dice is used? There are 6 different numbers (1 through 6) on dice, and in 1 roll of the dice, there is a 1-in-6 chance of getting a 6. Mathematically, it will be denoted as ⅙ or 0.16666. In percentage, that’s 16.67% - meaning a player getting 6 on a random roll of the dice is precisely that -  16.67%.

But what is the probability that it will rain tomorrow? Well, that’s far more complex, with way too many variables. And perhaps with information that is not always reliable. We can therefore see that we have absolute probability in some cases and in some only relative probability. To put that simply, we can be certain that there is a 50% chance of a coin yielding either heads or tails, but we’re not even 20% sure that there will be rain tomorrow.

Once you understand this basic fact, you’ll start to see how probability has permeated everyday life. For e.g., what’s the probability that a student is asked to join the Red house in a particular school, which has 4 houses - Blue, Red, Green & Yellow. The answer should be simple by now: there is definitely a 1-in-4 chance, meaning a 25% chance that the student ends up in the Red house.

Mathematical Formula for Probability

If the earlier examples of probability were clear, understanding the formula will become easier. To put it in simple terms: The Probability formula is one that gives us the possibility of an event occurring to the ratio of favorable outcomes.  

Probability that an event will occur P(E) = Number of favorable outcomes / Total number of outcomes

Here, favorable outcomes should not be confused with desirable outcomes.

So What is Joint Probability?

Now that we’ve gone through some scenarios to understand probability, we can delve a bit deeper. In the earlier example of a student randomly being assigned the Red house, we determined a 25% chance. But what if the question was modified slightly? In addition to finding out the probability that a student gets assigned to the Red house, we also want to know the chances of that student being put in the A division, out of 4 possible divisions - A, B, C & D.

This is what Joint Probability is all about. Not just fulfilling one condition, but two. Let’s look at another example.

Let’s say there is a bowl full of gems. Those sweet colorful candies that kids love. Let’s assume this bowl of gems filled with red, blue, yellow, green, and pink. So, if you were to dip your hand in this bowl and pick out just one candy, what is the probability that it will be blue? It’s going to be 1-in-5, as there are 5  candies and you have to put your hand in only once. But what is the probability that not only is the candy you pull out blue but also weighs more than 2 gms? For that, we’ll need to apply the formula of Joint Probability.

Formula for Joint Probability

The easiest way to represent joint probability is to assign letters A and B as the outcome of two events. From our earlier example, A can be the event of you pulling out one gems candy, and B will then be the event of the weight being more than or less than 2 gms. And if P is the probability of these events, then the joint probability will be calculated as:

P (A⋂B)

where A and B are 2 predefined events.

Here, the symbol represents the intersection of the events. Thus, the chances of event A and B happening at the same time is the intersection of A and B. This is why the intersection of both events happening at the same time is called joint probability.

Examples of Joint Probability

Let’s further try to express the joint probability examples with the help of mathematical terms. We can continue discussing the example of pulling out a blue-colored gem candy from a bowl that is filled with gems of all 5 colors: i.e., red, blue, yellow, green, and pink. But now, let’s assume that there are a total of 50 gems in the bowl, 10 of each color. And we’ve also taken the pain to weigh each gem individually and have the raw data for each.

So, let’s try to solve: What is the joint probability of pulling out a blue-colored gem from a bowl filled with 10 red, blue, yellow, green, and pink colored gems each, and that weighs equal or more than 2 gms.

In a tabular format, the data would look something like this

Colors

Weight (in gms)

Red

1.8

Red

2.1

Red

2.2

Red

1.9

Red

1.7

Red

1.6

Red

2

Red

2.2

Red

2.3

Red

1.8

Blue

2.1

Blue

2.2

Blue

1.9

Blue

1.7

Blue

1.8

Blue

2.2

Blue

2

Blue

1.6

Blue

1.7

Blue

1.9

Yellow

1.5

Yellow

1.7

Yellow

2.1

Yellow

1.8

Yellow

2.2

Yellow

2

Yellow

1.6

Yellow

1.8

Yellow

2.2

Yellow

2

Green

1.6

Green

2.1

Green

2

Green

1.9

Green

1.5

Green

1.6

Green

1.7

Green

1.8

Green

2.2

Green

2

Pink

1.6

Pink

2.2

Pink

1.9

Pink

2

Pink

2.1

Pink

1.8

Pink

2.2

Pink

2

Pink

1.6

Pink

1.5

Since the first event is whether or not a gem is Blue, that will be our first column. The row will contain information of the weight, whether or not the gem is more than 2gms.

Again, in a table format, it can be simplified as:

 

< 2 gms

>= 2gms

Blue

A

B

Not Blue

C

D

Here, we’re calculating the probabilities of 4 cells - A, B, C, and D - but the initial problem statement is only concerned with Blue gems whose eight equals or exceeds 2 gms, which is essentially cell B.

Based on the raw data available, we can quickly figure out the probability of:

Cell A: a gem picked at random is blue but less than 2 gms.

For this, firstly, the gem needs to be blue. The chances of that are 1/5.  Next, the blue gem’s weight should equal or exceed 2 gms. Looking at the raw data, we can observe that of all the 10 blue gems, only 4 equal or exceed 2 gms. Thus, for cell B, that would mean a 4-in-10 chance or 4/10= 0.4.

Thus, P (A⋂B) would yield 0.2 x 0.4, which is 0.08. This is a 8% chance that the gem you pull out randomly is blue and weighs at least 2 gms.

Maths Makes Data Analysis Easy

If your little one is still unsure how so much math is helpful in the future, you can point them towards Data Analysis - a field that is built on a clear understanding of the fundamentals of maths. And if the challenge right now is to understand maths, then data simplified with Cuemath will bring further clarity.

After all, the world is running on data, and most prominent tech companies hire specialist data experts to figure out various trends in a variety of businesses. All of which is possible if the basics of maths are clearly understood.

Happy learning!

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