Probability Distribution of a Discrete Random Variable

Statistics • Discrete Random Variables and Their Probability Distributions

Written by STEM Calculators Team Published December 13, 2025 Updated February 24, 2026

Probability Distribution of a Discrete Random Variable

Build a probability distribution for a discrete random variable x, verify the two required conditions, and visualize the distribution with a bar graph. A second visualization builds the classic two-stage tree diagram (two trials).

Core idea

Definition. A probability distribution of a discrete random variable lists all possible values the random variable can assume and their corresponding probabilities.

Two conditions (must always hold).

1) For each value of x, 0 ≤ P(x) ≤ 1

2) The sum of all probabilities is ΣP(x) = 1

Use the builder below to write the distribution as a table and verify these two conditions.

Distribution builder (table → probabilities)

Input mode Probabilities Frequencies

Value of x	Probability P(x)	Row

1) Probability distribution (output)

ΣP(x) = — Condition 1: — Condition 2: — Valid distribution: —

x	P(x)	Notes
Tip: if you entered frequencies, probabilities are computed as relative frequency = frequency ÷ total.

2) Read probabilities from the table

Choose an event

Event result

—

3) Visualizations

Bar graph of P(x)

Each bar height equals the probability for that value of x.

Tree diagram (two trials)

Build a two-stage probability tree: two selections, each either M (event happens) with probability 0.60 or N (event does not happen) with probability 0.40.

p = P(M) (default 0.60)

x = number of M outcomes	P(x)	How it is obtained
This matches the idea “count successes in 2 trials.”

Rate this calculator

0.0 /5 (0 ratings)

Be the first to rate.

Your rating

Name (optional) Review (optional)

You can update your rating any time.

Definition

A discrete random variable takes values in a countable set (for example, 0, 1, 2, 3, …). The probability distribution lists each possible value of x and the corresponding probability P(x).

Two required conditions

1) For each value of x, 0 ≤ P(x) ≤ 1

2) The sum of all probabilities is ΣP(x) = 1

If either condition fails, the table is not a valid probability distribution.

Example table (vehicles owned)

If x is the number of vehicles owned by a randomly selected family, a probability distribution can be written directly from relative frequencies (interpreted as probabilities).

x	P(x)	Reading from the table
0	0.015	P(x = 0) = 0.015
1	0.235	P(x = 1) = 0.235
2	0.425	P(x = 2) = 0.425
3	0.245	P(x = 3) = 0.245
4	0.080	P(x = 4) = 0.080
ΣP(x) = 1.000

Example of a “combined” event:
P(x > 2) = P(x = 3) + P(x = 4) = 0.245 + 0.080 = 0.325

Visualization

A discrete probability distribution is often shown with a bar graph where each bar height equals P(x). Another common visualization is a tree diagram for multi-stage experiments, where branch probabilities are multiplied along a path and then combined when outcomes represent the same value of x.

The calculator page draws both: a bar graph for your table and a two-stage tree diagram that produces a distribution for x.

Frequently Asked Questions

What is a probability distribution of a discrete random variable?

It is a list of all possible values of x and the probability assigned to each value P(x). The table represents how likely each outcome value is for the random variable.

What conditions must a discrete probability distribution satisfy?

Each probability must be between 0 and 1, and the total must add up to 1. In symbols: 0 <= P(x) <= 1 for every row and sum P(x) = 1.

How do you convert frequencies into probabilities?

Divide each frequency by the total frequency across all rows. This uses relative frequency: P(x) = frequency / total.

How do I compute P(x > a) or P(a < x < b) from a table?

Add the probabilities for all x values that satisfy the condition. For example, P(x > a) is the sum of P(x) for every row with x greater than a.

How does the two-stage tree diagram relate to a probability distribution?

In a two-stage process, branch probabilities are multiplied along each path and then combined for outcomes that produce the same x value. This creates a distribution for x such as the number of successes in 2 trials.