How to explain it
The anchor students hold onto: Uniform model: P = 1 / total outcomes for each outcome. Non-uniform model: P = observed frequency / total observations. In all models, the probabilities of every outcome must sum to 1.
You have modeled probability for individual events. Next, you will calculate probabilities for compound events — two or more outcomes combined — using lists, tables, and tree diagrams.
Worked examples
Example 1
Uniform Probability Model
Spinner: 8 sections, 2 red.
Step 1Sample space: 8 equal sections.
Step 2Uniform model: each section has P = 1/8.
Step 3Favorable: 2 red. P(red) = 2/8 = 1/4.
AnswerP(red) = 1/4
Example 2
Non-Uniform Model
20 spins: A=12, B=8.
Step 1Total observations: 20 spins.
Step 2Assign: P(A) = 12/20 = 3/5. P(B) = 8/20 = 2/5.
Step 3Verify: 3/5 + 2/5 = 5/5 = 1. Valid model.
AnswerP(A) = 3/5, P(B) = 2/5
Example 3
Compare & Explain
Model P(A)=1/3; 9 of 24 spins.
Step 1Model predicts: 1/3 × 24 = 8 expected.
Step 2Observed: 9 of 24 spins on A.
Step 3Compare: 9 vs. 8. Close — variation normal in small samples.
AnswerExpected: 8 · Observed: 9 · Variation expected.
Common mistakes
What students write
Assumes all outcomes are equally likely and assigns P = 1/n even when the problem is non-uniform or observed frequencies are unequal.
The fix
Check whether the problem provides a uniform model (equal sections, fair coin/die) or observed data. If data is given, build a non-uniform model: P = frequency / total for each outcome.
Try this
Liam is building a probability model for a spinner. He observes 50 spins: Red appeared 25 times, Blue appeared 15 times, Green appeared 10 times. Liam’s work: There are 3 colors, so each color is equally likely. P(Red) = 1/3, P(Blue) = 1/3, P(Green) = 1/3. Identify Liam’s error and build the correct probability model from the observed data.
What students write
Assigns the raw count as the probability (P = 8) instead of the relative frequency (P = 8/20), so the model sums to more than 1.
The fix
A valid probability must be between 0 and 1. Divide each count by the total number of observations to find relative frequency, then use that as the probability.
Try this
Maya observes 40 spins of a spinner and records: Section A appeared 8 times, Section B appeared 16 times, Section C appeared 16 times. Maya’s work: P(A) = 8, P(B) = 16, P(C) = 16 Maya says this is a valid model because she used the actual data. Identify Maya’s error and build the correct probability model.
Teacher tip
Head off the two predictable errors before they happen. First: Check whether the problem provides a uniform model (equal sections, fair coin/die) or observed data. If data is given, build a non-uniform model: P = frequency / total for each outcome. Second: A valid probability must be between 0 and 1. Divide each count by the total number of observations to find relative frequency, then use that as the probability.