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Uniform
Probability Models

Learn how to assign equal probability to every outcome and use it to find the probability of any event.

7th Grade Math

What Is Probability?

Probability measures how likely an event is to occur. It is always a number from \(0\) to \(1\), where \(0\) means impossible and \(1\) means certain.

\(P(E) = 0\)
Impossible: it will never happen
\(P(E) = 0.5\)
Equally likely: 50-50 chance
\(P(E) = 1\)
Certain: it will always happen

Key Vocabulary

Experiment
An action with uncertain results: flipping a coin, rolling a die
Outcome
One possible result of the experiment: getting "Heads"
Sample Space
The set of all possible outcomes: \(\{H, T\}\) for a coin
Event
A subset of the sample space you care about: rolling an even number

Why Uniform?

A probability model is called uniform when every single outcome is equally likely. This is the simplest and most common type of probability model.

1 2 3 4 5 6
\(\frac{1}{6}\)
 P(outcome)

The Uniform Probability Formula

To find the probability of an event (a group of outcomes), count how many outcomes are in the event and divide by \(n\):

Visualizing \(\frac{|E|}{n}\)

Sample Space: \(n = 6\)
 1 2 3 4 5 6
Event "even number": \(|E| = 3\) highlighted in green

Uniform vs Non-Uniform

Uniform Model Non-Uniform Model
Outcome Probabilities All equal Can be different
Example Fair coin, fair die Loaded die, weighted spinner
Formula \(P = \frac{1}{n}\) for each Each outcome assigned individually
Sum of All \(= 1\) \(= 1\)
Complexity Simpler to compute Requires more information

How to Apply the Model

1
Identify the experiment. What random process is happening?
2
List all outcomes. Write the complete sample space and count \(n\).
3
Verify equally likely. Confirm each outcome has the same chance (fair device, random draw, etc.).
4
Define the event. Identify which outcomes count as "favorable."
5
Calculate. Use \(P(E) = \frac{|E|}{n}\) and simplify.

Example 1: Fair Coin

A fair coin is flipped once. What is the probability of landing on heads?

H Heads T Tails
1
Sample space: \(\{H, T\}\), so \(n = 2\).
2
The event "heads" has 1 favorable outcome.
3
\(P(\text{heads}) = \frac{1}{2} = 0.5\).

Example 2: Fair Spinner

A spinner is divided into 8 equal sections numbered 1 through 8. What is the probability of landing on a number greater than \(5\)?

1 2 3 4 5 6 7 8

Example 3: Drawing a Card

A standard deck has 52 cards. You draw one card at random. What is the probability of drawing a heart?

♥ Hearts 13 cards ♦ Diamonds 13 cards ♣ Clubs 13 cards ♠ Spades 13 cards

Example 4: Letters in a Bag

The letters M, A, T, H, S are each written on separate tiles and placed in a bag. One tile is drawn at random. Find the probability of each event.

M A T H S
1
Drawing a vowel: Only A is a vowel. \(P(\text{vowel}) = \frac{1}{5}\).
2
Drawing a consonant: \(\{M, T, H, S\}\) so \(P(\text{consonant}) = \frac{4}{5}\).
3
Drawing M or T: Two favorable outcomes. \(P(M \text{ or } T) = \frac{2}{5}\).

Placing Probabilities on a Number Line

0
\(\frac{1}{4}\)
\(\frac{1}{2}\)
\(\frac{3}{4}\)
 1 Impossible Equally Likely Certain
\(P(\text{heads}) = \frac{1}{2}\)
\(P(\text{heart}) = \frac{1}{4}\)
\(P(\text{consonant}) = \frac{4}{5}\)

Probabilities Always Sum to 1

In any probability model, the probabilities of all individual outcomes must add up to exactly \(1\). In a uniform model, this check is simple:

This is a powerful error-checking tool. If your probabilities do not add to \(1\), something is wrong: you may have missed an outcome or miscounted.

Common Mistakes

✕ Assuming Uniform
Not all situations are equally likely. A spinner with unequal sections is not uniform.
✕ Forgetting Outcomes
Miscounting the sample space (e.g., forgetting "face cards" include Jacks, Queens, and Kings).
✕ Wrong Denominator
Using the number of favorable outcomes as the denominator instead of the total number of outcomes.

Summary of Key Ideas

Concept Key Fact
Uniform Model All outcomes are equally likely
Individual Probability \(P(\text{outcome}) = \frac{1}{n}\)
Event Probability \(P(E) = \frac{|E|}{n}\)
Sum Rule All probabilities add to 1
Range \(0 \le P(E) \le 1\)
Complement \(P(\text{not } E) = 1 - P(E)\)

Question 1 of 5

A bag contains 10 marbles, each a different color. You draw one marble at random. What is the probability of drawing the red marble?

Question 2 of 5

A fair six-sided die is rolled once. What is the probability of rolling a number less than 3?

Question 3 of 5

A spinner has 5 equal sections labeled A, B, C, D, E. What is the probability of not landing on B?

Question 4 of 5

Which of the following is not a uniform probability model?

Question 5 of 5

A standard deck of 52 cards has 4 aces. If you draw one card at random, what is \(P(\text{ace})\)?