Probability Mass Function Explained: Formula, Properties, and Examples

The concept of Probability Mass Function plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Understanding how a Probability Mass Function works is essential for students preparing for competitive exams, as well as anyone seeking to master statistics and probability topics.

What Is Probability Mass Function?

A Probability Mass Function (PMF) is a mathematical function that gives the probability that a discrete random variable is exactly equal to a particular value. You’ll find this concept applied in areas such as binomial and Poisson distributions, statistics, and data analysis. The PMF helps describe the behavior of discrete outcomes—like rolling dice, flipping coins, or counting the number of cars passing a point in an hour.

Key Formula for Probability Mass Function

Here’s the standard formula:

$$P(X = x) = f(x)$$

This means the probability that a discrete random variable

$$X$$ takes the value $$x$$ is represented by the function $$f(x)$$ . Every possible value of $$x$$ has a probability assigned by this function.

Step-by-Step Illustration

Let’s see how to find the Probability Mass Function for a simple example:

Value of X (x)	P(X = x)
0	0.1
1	0.2
2	0.3
3	0.4

Notice that the sum of all probabilities is 1:

$$0.1 + 0.2 + 0.3 + 0.4 = 1$$

To solve a PMF problem, follow these steps:

1. Identify all possible values of the discrete variable.

2. Assign probabilities to each outcome so that each is between 0 and 1.

3. Ensure the sum of these probabilities equals 1.

Common Exam PMF Applications (Binomial and Poisson)

Binomial and Poisson distributions are two important cases for PMF. Here’s what the probability mass function looks like for each:

Distribution	PMF Formula	Typical Example
Binomial	$$P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$$	Number of heads in 10 coin tosses
Poisson	$$P(X = k) = \dfrac{e^{-\lambda}\lambda^{k}}{k!}$$	Number of phone calls in a minute

Properties and When to Use a PMF

• The probability of every possible value is between 0 and 1.
• The sum of all probabilities over all possible values is always 1.
• PMF is used for discrete (separate/countable) random variables, not continuous ones.
• No two values can get the same exact probability unless allowed by the variable’s definition.

Difference: PMF vs PDF vs CDF

Aspect	PMF	PDF	CDF
Type of Variable	Discrete	Continuous	Both
Expression	Probability of each exact value	Probability over an interval (not single value)	Probability up to a certain value (cumulative)

Practice Problem: Find the PMF Constant

Example: Let the PMF be

$$f(x) = k, 2k, 3k$$ for values $$x=1,2,3$$ , and $$f(x) = 0$$ elsewhere. Find k.

1. List all probabilities: $$f(1) = k,\ f(2) = 2k,\ f(3) = 3k$$ .

2. The sum must be 1: $$k + 2k + 3k = 1 \Rightarrow 6k = 1$$

3. Solve for k: $$k = \dfrac{1}{6}$$

So the PMF is:

x	P(X = x)
1	1/6
2	2/6 = 1/3
3	3/6 = 1/2

Real Life Examples of Probability Mass Function

• Counting the number of customers entering a shop in an hour.
• Rolling a die—PMF assigns 1/6 probability to each result from 1 to 6.
• Quality inspection: Probability that a batch has 0, 1, 2, or 3 defective items.

Cross-Disciplinary Usage

Probability Mass Function is not only useful in Maths but also plays an important role in Physics, Computer Science, and daily logical reasoning. Students preparing for JEE or NEET will see its relevance in various questions, including core probability problems and discrete probability distributions.

Try These Yourself

• Write the PMF for tossing two coins.
• For a six-sided dice, what is the PMF for getting even numbers?
• Check if the following is a valid PMF: 0.2, 0.5, 0.3 for three possible values.
• Find the probability a random variable equals 0, if PMF is $$1/4, 1/2, 1/4$$ for 0, 1, 2.

Frequent Errors and Misunderstandings

• Confusing PMF (for discrete) with PDF (for continuous variables).
• Assigning probabilities that do not sum to 1—always check your total!
• Assigning negative or greater than 1 values as probabilities.
• Forgetting PMF only applies to countable outcomes—not measurements or continuous ranges.

Relation to Other Concepts

The idea of Probability Mass Function connects closely with topics such as Probability Distribution and Cumulative Distribution Function (CDF). Mastering this helps with understanding more advanced concepts in future chapters, like random variables and statistical inference.

Classroom Tip

A quick way to remember Probability Mass Function is: “Mass” means it gives the ‘weight’ (probability) attached to each individual possible value. Vedantu’s teachers often use a table or bar chart to visually show the PMF for dice or coin examples in live online classes.

We explored Probability Mass Function—from definition, formula, examples, mistakes, and connections to other subjects. Continue practicing with Vedantu to become confident in solving problems using this concept. For more detailed guidance, check out related topics like Random Variables.