Introduction to Probability Distributions

Learn what probability distributions are and how they describe the likelihood of different outcomes.

Advanced25 minLesson
Printable worksheet

Definition

A probability distribution describes how likely each possible outcome is for a random experiment or variable.
For a discrete probability distribution:
  • List all possible outcomes
  • Assign a probability to each outcome
  • All probabilities must be between 0 and 1
  • The sum of all probabilities must equal 1
∑i=1nP(xi)=1\sum_{i=1}^{n} P(x_i) = 1
A probability distribution can be shown as a table, formula, or graph (called a probability histogram).

Try it now

For a valid probability distribution, all probabilities must sum to:

Worked Examples

Create a probability distribution for rolling a single fair die.
1

Identify all possible outcomes

A die can show 1, 2, 3, 4, 5, or 6 → 6 outcomes
2

Determine probability of each outcome

Fair die means each face is equally likely: P=16P = \frac{1}{6} → P(x)=16P(x) = \frac{1}{6} for each
3

Verify probabilities sum to 1

16+16+16+16+16+16=66=1\frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6} = \frac{6}{6} = 1 → Valid distribution
4

Write the distribution table

| xx | 1 | 2 | 3 | 4 | 5 | 6 | | P(x)P(x) | 16\frac{1}{6} | 16\frac{1}{6} | 16\frac{1}{6} | 16\frac{1}{6} | 16\frac{1}{6} | 16\frac{1}{6} | → Uniform distribution

Common Mistakes

Probabilities that do not sum to 1

Why it's wrong: A valid probability distribution MUST have all probabilities sum to exactly 1. If they do not, you have either missed an outcome or made a calculation error.

Correct: Always verify: ∑P(xi)=1\sum P(x_i) = 1. If the sum is less than 1, you are missing outcomes. If greater, you have counted something twice.

Confusing probability with frequency

Why it's wrong: Frequency is a count (e.g., '6 ways to roll a 7'), while probability is a ratio (e.g., 'P(7)=6/36P(7) = 6/36').

Correct: Probability = favorable outcomes divided by total outcomes. Always express as a fraction, decimal, or percentage.

Assuming all outcomes are equally likely

Why it's wrong: Many distributions are NOT uniform. For two dice, sum of 7 is more likely than sum of 2.

Correct: Carefully count the ways to achieve each outcome. Only simple experiments (single die, coin flip) tend to be uniform.

Interactive Visual

Bar Chart

Sum of two dice

2(0.03)
3(0.06)
4(0.08)
5(0.11)
6(0.14)
7(0.17)
8(0.14)
9(0.11)
10(0.08)
11(0.06)
12(0.03)

Dice Roller

Click the button to run trials and see probability in action!

Interactive Sandbox

Expression Calculator

Try these:

History

No calculations yet

Practice Problems

15 problems
Problem 1 of 15
Easy
For a valid probability distribution, all probabilities must sum to:

Why It Matters

Probability distributions are the foundation of statistics and data science:
  • Quality Control: Manufacturers use distributions to predict defect rates
  • Insurance: Companies calculate premiums based on risk distributions
  • Gaming: Casinos design games using probability distributions to ensure profit
  • Weather: Meteorologists model temperature and rainfall using distributions
  • Medicine: Clinical trials analyze drug effectiveness using distributions
Understanding distributions helps you make informed decisions in uncertain situations!

Real World Applications

Quality Control in Manufacturing

Factories use probability distributions to predict how many defective items will be produced.

Example:

If a machine produces 1000 items and has a 2% defect rate, the expected number of defects is 1000×0.02=201000 \times 0.02 = 20.

1Try It Yourself

A factory produces light bulbs. Each bulb has a 3% chance of being defective. In a batch of 500 bulbs:

What is the expected number of defective bulbs?

Step 1: Write the mathematical expression

Calculate: 500×0.03500 \times 0.03

Insurance Risk Assessment

Insurance companies use probability distributions to calculate fair premiums based on claim likelihood.

Example:

If 5% of drivers file claims averaging 5000 euros, the expected payout per driver is 0.05×5000=2500.05 \times 5000 = 250 euros.

2Try It Yourself

A health insurance company knows that 8% of customers file claims averaging 2000 euros. They have 10000 customers.

What is their expected total payout?

Step 1: Write the mathematical expression

Calculate: 10000×0.08×200010000 \times 0.08 \times 2000

Key Takeaways

  • 1A probability distribution shows the probability of each possible outcome
  • 2All probabilities must be between 0 and 1, and must sum to exactly 1
  • 3A probability histogram displays the distribution visually with bars
  • 4Expected value E(X)=∑xi⋅P(xi)E(X) = \sum x_i \cdot P(x_i) gives the long-run average outcome
  • 5Not all distributions are uniform - some outcomes can be more likely than others

Frequently Asked Questions

Discrete distributions have countable outcomes (like dice rolls: 1, 2, 3, 4, 5, 6). Continuous distributions have uncountable outcomes (like height, which can be any value in a range). This lesson focuses on discrete distributions.
Discrete distributions have countable outcomes (like dice rolls: 1, 2, 3, 4, 5, 6). Continuous distributions have uncountable outcomes (like height, which can be any value in a range). This lesson focuses on discrete distributions.
No! Probabilities are always between 0 and 1 (inclusive). A probability of 0 means impossible, and 1 means certain.
Expected value is the average outcome if you repeated the experiment many times. It does not mean you will get that exact value - it is a long-run average.

Glossary

Probability distribution
A complete description of all possible outcomes and their probabilities
Random variable
A variable whose value is determined by chance (e.g., the number shown on a die)
Discrete distribution
A distribution where outcomes are countable (integers, categories)
Expected value
The weighted average of all possible outcomes: E(X)=∑xi⋅P(xi)E(X) = \sum x_i \cdot P(x_i)
Probability histogram
A bar graph showing probabilities of each outcome
Uniform distribution
A distribution where all outcomes have equal probability

Formula Card

Probability Sum Rule

∑i=1nP(xi)=1\sum_{i=1}^{n} P(x_i) = 1

All probabilities must sum to 1

Expected Value

E(X)=∑i=1nxi⋅P(xi)E(X) = \sum_{i=1}^{n} x_i \cdot P(x_i)

Weighted average of outcomes

Variance

Var(X)=∑i=1n(xi−μ)2⋅P(xi)\text{Var}(X) = \sum_{i=1}^{n} (x_i - \mu)^2 \cdot P(x_i)

Measure of spread around the mean

Today's challengeOne problem, three tries

More in This Topic

Binomial Probability

Advanced25 min

Discrete vs Continuous Distributions

Advanced25 min

Expected Value

Advanced25 min

Introduction to Normal Distribution

Advanced25 min