What is a Probability Mass Function?

S7-SA3-0027

Grade Level:

Class 12

AI/ML, Physics, Biotechnology, FinTech, EVs, Space Technology, Climate Science, Blockchain, Medicine, Engineering, Law, Economics

Definition

What is it?

A Probability Mass Function (PMF) tells us the probability for each possible outcome of a discrete random variable. Think of it as a special rule or formula that assigns a probability (a number between 0 and 1) to every single value a variable can take.

Simple Example

Quick Example

Imagine you roll a standard six-sided dice. The possible outcomes are 1, 2, 3, 4, 5, 6. A PMF would tell you the probability of rolling a 1 (which is 1/6), the probability of rolling a 2 (also 1/6), and so on for all numbers. It clearly lists the chance for each specific result.

Worked Example

Step-by-Step

Let's say a local shop sells samosas, and the number of samosas a customer buys can be 0, 1, or 2. We want to find the PMF.

Step 1: List all possible outcomes for the number of samosas bought (X). So, X can be 0, 1, or 2.

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Step 2: Collect data on past customers. Out of 100 customers, 20 bought 0 samosas, 50 bought 1 samosa, and 30 bought 2 samosas.

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Step 3: Calculate the probability for each outcome.
- P(X=0) = (Number of customers who bought 0) / (Total customers) = 20/100 = 0.2

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Step 4: Calculate P(X=1).
- P(X=1) = (Number of customers who bought 1) / (Total customers) = 50/100 = 0.5

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Step 5: Calculate P(X=2).
- P(X=2) = (Number of customers who bought 2) / (Total customers) = 30/100 = 0.3

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Step 6: Write down the PMF. It's a list or table pairing each outcome with its probability.
- P(X=0) = 0.2
- P(X=1) = 0.5
- P(X=2) = 0.3

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Step 7: Check if the sum of probabilities is 1. 0.2 + 0.5 + 0.3 = 1. Yes, it is. This is our PMF.

Answer: The PMF is P(X=0)=0.2, P(X=1)=0.5, P(X=2)=0.3.

Why It Matters

PMFs are super important in fields like AI/ML to predict events, in FinTech to assess risks, and in Medicine to understand disease spread. Knowing PMFs helps data scientists, engineers, and doctors make better decisions and understand uncertain situations.

Common Mistakes

MISTAKE: Thinking PMF values can be greater than 1 or negative. | CORRECTION: The probability for any single outcome must always be between 0 and 1 (inclusive). You can't have a 120% chance or a -10% chance.

MISTAKE: Not checking if all probabilities in a PMF add up to 1. | CORRECTION: The sum of all probabilities for all possible outcomes in a PMF must always equal exactly 1. If it doesn't, something is wrong.

MISTAKE: Using PMF for continuous variables (like height or weight). | CORRECTION: PMFs are only for discrete random variables, which have outcomes that can be counted (like number of heads, number of students). For continuous variables, you use a Probability Density Function (PDF).

Practice Questions

Try It Yourself

QUESTION: A spinner has 3 equal sections: Red, Blue, Green. What is the PMF for the color it lands on? | ANSWER: P(Red) = 1/3, P(Blue) = 1/3, P(Green) = 1/3

QUESTION: A bag contains 5 red balls and 3 blue balls. You pick one ball. Let X be 1 if it's red and 0 if it's blue. What is the PMF for X? | ANSWER: P(X=1) = 5/8, P(X=0) = 3/8

QUESTION: A coin is tossed two times. Let Y be the number of heads. List the possible outcomes (HH, HT, TH, TT) and then find the PMF for Y. | ANSWER: Possible outcomes for Y are 0, 1, 2. P(Y=0) = 1/4 (for TT), P(Y=1) = 2/4 = 1/2 (for HT, TH), P(Y=2) = 1/4 (for HH).

MCQ

Quick Quiz

Which of the following is a necessary property of a Probability Mass Function P(x) for a discrete random variable X?

P(x) can be negative for some values of x

The sum of all P(x) values must be less than 1

P(x) must be between 0 and 1 (inclusive) for all x, and the sum of all P(x) must be 1

P(x) is used for continuous random variables

The Correct Answer Is:

Option C correctly states the two fundamental properties of a PMF: individual probabilities must be between 0 and 1, and their sum over all possible outcomes must equal 1. Options A, B, and D describe incorrect properties.

Real World Connection

In the Real World

In cricket analytics, PMFs can be used to model the number of wickets a bowler might take in a match or the number of runs a batsman might score in an over. This helps teams make strategic decisions, like who to bowl next or how aggressive to play, using data instead of just guesswork.

Key Vocabulary

Key Terms

DISCRETE RANDOM VARIABLE: A variable whose values can be counted (like 0, 1, 2, 3). | PROBABILITY: The chance of an event happening, a number between 0 and 1. | OUTCOME: A possible result of an experiment or event. | SUMMATION: Adding up all values in a set. | EVENT: A specific set of outcomes.

What's Next

What to Learn Next

Now that you understand PMFs, explore 'Expected Value' and 'Variance' of discrete random variables. These concepts build on PMFs to help you understand the average outcome and how spread out the outcomes are, which is super useful for predictions!