Grade 7 · Grade 7: Sampling and Probability · Sampling and Probability
Probability Models and Compound Events
Move between theoretical and experimental probability, then count compound outcomes with lists, tables and tree diagrams.
Learning objectives
- Place probabilities on a 0 to 1 scale.
- Compare experimental relative frequency with theoretical probability.
- List sample spaces for compound events.
- Use simulation to estimate probabilities.
AERO Mathematics alignment
AERO.M7.SP.4
Understand that the probability of a chance event is a number between 0 and 1 that expresses its likelihood.
AERO.M7.SP.5
Approximate the probability of a chance event by collecting data, and compare experimental results with a theoretical probability model.
AERO.M7.SP.6
Find probabilities of compound events using organised lists, tables, tree diagrams and simulation.
MYP criteria
- Criterion A — Knowing and understanding
- Criterion B — Investigating patterns
- Criterion C — Communicating
- Criterion D — Applying mathematics in real-life contexts
ENGAGE
Start here
You toss a fair coin nine times and get heads every time.
Think about it
What is the probability the tenth toss is heads?
Hint: Does the coin remember what it did before?
EXPLAIN
From theory to experiment
The probability of an event is a number from 0 (impossible) to 1 (certain). Theoretical probability uses the structure of the situation: a fair die gives P(4) = 1/6. Experimental probability uses data: relative frequency = successes ÷ trials.
| Trials | Heads | Relative frequency |
|---|---|---|
| 10 | 7 | 0.70 |
| 100 | 54 | 0.54 |
| 1000 | 508 | 0.508 |
As the number of trials grows, the relative frequency settles near the theoretical value. This is the law of large numbers, and it says nothing at all about the next single trial.
Compound events
For two coins, the sample space is HH, HT, TH, TT, so P(exactly one head) = 2/4 = 0.5. Organised lists, tables and tree diagrams make sure no outcome is missed.
Common misconception
There is no law of averages for a fair coin. After nine heads, the probability of heads next is still exactly 0.5, because tosses are independent.
INVESTIGATION
Try it yourself
Roll two dice 50 times and record the sums. Compare your experimental probabilities with the theoretical ones from a 6 × 6 table. Which sum was furthest from theory, and what would you expect after 500 rolls?
Watch
Basic Probability
Math Antics · 11:28
Connects theoretical probability with what actually happens when you experiment.
Before you watch: Can a probability be 1.4?
- Estimate P(heads) after 500 tosses. How close is it to 0.5?
- List the sample space for tossing two coins.
Interactive simulation · PhET
Plinko Probability
Compare the experimental distribution with the theoretical one as the number of trials grows.
While you explore
- How close is the experimental probability to the theoretical value after 1000 trials?
- What does this tell you about a single trial?
Key vocabulary
- Relative frequency
- How often something actually happened, as a fraction of trials.
- Sample space
- A list of everything that could happen.
Practice questions
0/1 correct
Level 1 · Criterion A
A fair six-sided die is rolled. What is P(rolling a number greater than 4)?
MYP criterion tasks
Level 2 · Criterion B
Two coins are tossed. List the sample space and find P(at least one head).
Reflect & track
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