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Class 12 Statistics Notes · GSEB

Discrete Probability Distribution

Discrete Probability Distribution — learn to construct and verify probability distribution tables using the two essential conditions. GSEB Class 12 Commerce Statistics notes with interactive examples.

Last updated: 22 Sep 2026

Notes

What is a Probability Distribution?

Probability Distribution
A probability distribution is a table, formula, or graph that lists all possible values of a random variable X along with their corresponding probabilities P(X = x). It completely describes the behaviour of a random variable.
Think of it as a complete "scoreboard" — every possible outcome gets its probability score, and all scores add up to 1 (certainty).

The Two Essential Conditions

For any valid probability distribution, two conditions must hold. If either fails, the table is not a valid distribution.

Conditions for a Valid Distribution

P(x)≥0 for all xand∑P(x)=1P(x) \geq 0 \text{ for all } x \quad \text{and} \quad \sum P(x) = 1

Condition 1: P(x) ≥ 0

Every probability must be non-negative. ✓ Satisfied

Condition 2: ΣP(x) = 1

Sum of all probabilities must equal 1. Current sum = 1.0000. ✓ Satisfied

Explore: Interactive Distribution Tables

Select a preset distribution below. The table, bar chart, and condition checker update in real time.

About this distribution

X = number of heads in 2 fair coin tosses. P(0) = 1/4, P(1) = 2/4, P(2) = 1/4.
x012Σ
P(X = x)0.25000.50000.25001.0000
Probability Bar Chart
0.25
0
0.50
1
0.25
2

x values

Solved Examples

Solved Example

Problem

A random variable X has the following distribution: X: 0 1 2 P: 0.3 k 0.5 Find the value of k.

Solution

k = 0.2

Solved Example

Problem

Check whether the following is a valid probability distribution: X: 1 2 3 4 P: 0.2 0.3 0.4 0.2

Solution

Not a valid distribution (sum = 1.1 ≠ 1)

Solved Example

Problem

Two coins are tossed. Let X = number of heads. Write the probability distribution of X.

Solution

P(X=0) = 0.25, P(X=1) = 0.50, P(X=2) = 0.25

Key Takeaways

Key Takeaways

  • A probability distribution lists all possible values of X and their probabilities P(X = x).
  • Two mandatory conditions: (1) P(x) ≥ 0 for all x, and (2) ΣP(x) = 1.
  • The sum of all probabilities always equals 1 — this represents total certainty.
  • Discrete distributions are shown as tables or bar charts; continuous distributions use density curves.
  • Always verify both conditions when given a distribution table in exams.

Practice

  1. A random variable X has distribution: P(X=1) = 0.2, P(X=2) = 0.5, P(X=3) = k. Find k.
  2. Is the following a valid distribution? X: 1, 2, 3 with P: 0.4, 0.4, 0.4.
  3. Three coins are tossed. Let X = number of heads. Write the probability distribution of X.

Explore Further