Probability Models: Uniform and Non-Uniform

Lesson 1 of 1

In this lesson you will:

• Define probability models and their requirements
• Build uniform and non-uniform models
• Compare model predictions to observed data

Learning Objectives for This Lesson

By the end of this lesson, you will be able to:

1. Define a probability model as an assignment of probabilities to all outcomes
2. State the two requirements of a valid probability model
3. Develop a uniform probability model for equally likely outcomes
4. Use a uniform model to compute the probability of an event
5. Build a non-uniform probability model from observed frequency data
6. Compare modeled probabilities to observed frequencies and explain discrepancies

"Can You Really Trust One-Sixth?"

When you roll a fair die, you say .

But how do you know that's right?

• From 7.SP.C.6: experimental probability approaches theoretical with more trials
• So where does come from mathematically?
• Today we build the formal framework: the probability model

What a Probability Model Contains

A probability model has two parts:

• Sample space — the complete set of all possible outcomes
• Probability assignment — a number between 0 and 1 for each outcome

Die example: sample space ; each face gets a probability. Every outcome must be included.

Die Model: Outcomes and Probabilities

• Every outcome from 1 to 6 is listed — sample space is complete
• Each probability equals — a number between 0 and 1
• Sum: — the requirement is satisfied

Two Requirements for a Valid Model

Every probability model must satisfy both conditions:

Condition 1: Each probability is between 0 and 1 (inclusive)

Condition 2: All probabilities sum to exactly 1

Missing any outcome means the sum won't reach 1 — the model is incomplete.

Valid or Invalid? Four Candidate Models

Both invalid models fail Condition 2. The die and A/B/C models satisfy both conditions.

Check-In: Is This a Valid Probability Model?

A student proposes this model for a spinner with 4 sections:

Questions:

• Does each probability satisfy Condition 1?
• Does the sum satisfy Condition 2?
• Is this a valid probability model?

From Definition to Application: Uniform Models

We know what a probability model is.

Now: how do we assign probabilities?

• Simplest case: all outcomes are equally likely
• This is the uniform probability model
• Formula coming up — it builds directly on the die example

Uniform Models: When and How to Apply

Use when: all outcomes are equally likely — if outcomes, each gets

Always check: Are all outcomes equally likely?

Uniform Model: Visualizing Event Probability

Formula Card: Uniform Probability Model

Four-step process:

1. Identify the sample space (list all outcomes)
2. Confirm all outcomes are equally likely
3. Count favorable outcomes (those in the event)
4. Divide favorable by total

Worked Example: Prime Number on a Die

Event: rolling a prime on a fair six-sided die

• Sample space: — equally likely ✓
• Favorable: — 1 is not prime

Worked Example: Face Card from a Deck

Event: drawing a face card (J, Q, K) from a 52-card deck

• Sample space: 52 cards, equally likely ✓
• Favorable: 3 face ranks × 4 suits = 12 face cards

Guided Practice: Vowel from the Alphabet

Event: selecting a random letter from the alphabet and getting a vowel

Sample space: — 26 letters, equally likely ✓

Your turn:

• Count the favorable outcomes (vowels: A, E, I, O, U)
• Compute as a fraction and decimal

Check-In: Even Number on a Die

A fair die is rolled. Find using all four steps:

1. Sample space:
2. Uniform model? (fair die — yes)
3. Favorable outcomes:

What If Outcomes Are NOT Equally Likely?

The uniform model only works when all outcomes are equally likely.

Real-world examples where it fails:

• A thumbtack: lands point-up more often than point-down
• A spinner with unequal sections
• A loaded die that favors certain faces

We need a different approach — the non-uniform probability model.

Non-Uniform Probability Models: The Idea

A non-uniform probability model assigns different probabilities to outcomes.

• The two validity conditions still apply (each P in [0,1]; sum = 1)
• Probabilities are not equal — built from observed frequency data
• More data → more reliable estimates

Three Steps to Build a Non-Uniform Model

Step 1: Collect data — record outcome frequencies over many trials

Step 2: Compute relative frequencies — divide each count by the total

Step 3: Use relative frequencies as probability estimates; verify sum ≈ 1

Probabilities can be decimals (e.g., 0.35) — exact fractions not required.

Spinner Data Becomes a Probability Model

• Data from 200 spins converted to relative frequencies
• Each relative frequency becomes a probability estimate
• Sum check: ✓

Worked Example: Building a Loaded Die Model

A die is rolled 120 times. Frequencies: 1→15, 2→18, 3→25, 4→20, 5→22, 6→20.

Sum = 1.000 ✓ — compare to uniform: each would be . Face 3 is favored.

Using the Loaded Die Model for Predictions

Event: rolling a number ≥ 5 (outcomes {5, 6})

Non-uniform model:

Uniform model (if die were fair):

The models give different answers because the die is biased.

Guided Practice: Build a Spinner Model

Spinner spun 200 times — Red: 70, Blue: 90, Yellow: 40

Your tasks:

1. Convert each count to a relative frequency (÷ 200)
2. Verify sum = 1
3. Compute
4. Compare to uniform model:

Check-In: When to Use a Non-Uniform Model

Which of these situations requires a non-uniform probability model?

A) Rolling a fair six-sided die and finding

B) Spinning a penny on a table and finding

C) Drawing a card from a well-shuffled standard deck

D) Selecting a student at random from an evenly-sized class roster

How Well Does Your Model Fit Reality?

We can build models. Now: how do we know if a model is good?

• A model predicts expected frequencies:
• Actual observations differ from predictions
• Comparison tells us: trust, investigate, or revise

Comparing Predicted to Observed Frequencies

Predicted frequency = probability × number of trials

Interpreting differences:

• Small differences → model probably fits
• Large differences → investigate: wrong model or random variation?
• More trials makes large differences more meaningful

Worked Example: Spinner Model Fits Well

Model: , , — 100 spins.

→ Model fits well.

Worked Example: Spinner Model Fits Poorly

Same model — 100 spins, different trial.

→ Investigate.

Sample Size Changes How We Interpret Differences

• 60 rolls: max difference = 3 → within normal variation
• 600 rolls: max difference = 30 (30% deviation) → likely model mismatch
• Rule: more trials make large differences more meaningful

Guided Practice: Write a Model Verdict

Fair die claimed. In 60 rolls:

Fair model predicts 10 per face

Tasks:

1. Find (observed − predicted) for each face
2. Identify the maximum absolute difference
3. Write a verdict: fits well, poorly, or uncertain?

Key Takeaways from Today's Lesson

A probability model = sample space + probability assignment (both conditions required).

Uniform model: use when outcomes are equally likely; .

Non-uniform model: build from data when outcomes are not equally likely.

Five warnings:

• Every outcome must be listed — missing any makes the model incomplete
• Always ask "equally likely?" before using the uniform formula
• Probabilities can be decimals — don't force fractions that don't reduce
• The model predicts long-run behavior, not exact single-trial outcomes
• Non-uniform models need large samples to be reliable

Preview: Compound Events and Sample Spaces

Next lesson: 7.SP.C.8 — extend today's models to multi-step processes.

Coming up: compound events

• Drawing two cards in sequence
• Rolling two dice simultaneously
• Flipping a coin and rolling a die

Today's sample-space thinking is the foundation.