Grade 10 · Grade 10: Probability Rules and Distributions · Probability, Counting and Distributions
Probability Rules, Counting and the Normal Distribution
Combine probabilities with the addition and multiplication rules, count outcomes, and model data with the normal curve.
Learning objectives
- Apply the addition and multiplication rules, including for independent and dependent events.
- Use permutations, combinations and the counting principle.
- Evaluate study design, sampling methods and bias.
- Use mean and standard deviation with the normal distribution.
AERO Mathematics alignment
AERO.M10.SP.1
Apply the addition and multiplication rules of probability, including mutually exclusive, independent and dependent events, using Venn diagrams and tree diagrams.
AERO.M10.SP.2
Use permutations and combinations, together with the fundamental counting principle, to compute probabilities of compound events.
AERO.M10.SP.3
Distinguish between surveys, experiments and observational studies, evaluate sampling methods and bias, and communicate findings from a complete statistical investigation.
AERO.M10.SP.4
Use mean and standard deviation to describe data, and use the normal distribution to estimate proportions of populations where appropriate.
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
A quality inspector finds that 2% of items are faulty and picks two items at random.
Think about it
What is the chance that both are faulty, and does the first pick change the second?
Hint: Ask whether the events are independent.
EXPLAIN
Rules, counting and the normal curve
Combining probabilities
| Rule | Formula | Condition |
|---|---|---|
| Addition | P(A or B) = P(A) + P(B) − P(A and B) | Always |
| Mutually exclusive | P(A or B) = P(A) + P(B) | Events cannot both happen |
| Multiplication | P(A and B) = P(A) × P(B | A) | Always |
| Independent | P(A and B) = P(A) × P(B) | One does not affect the other |
Counting outcomes
The fundamental counting principle multiplies the choices at each stage. Permutations count arrangements where order matters, nPr = n!/(n − r)!, while combinations count selections where it does not, nCr = n!/(r!(n − r)!). Choosing 3 students from 10 gives 10C3 = 120 ways.
Study design
A survey describes a population, an observational study looks for association, and only a controlled experiment with random assignment supports a causal claim. Watch for selection bias, non-response bias and leading questions.
The normal distribution
Many natural measurements cluster symmetrically about the mean. About 68% of values lie within one standard deviation, about 95% within two and about 99.7% within three.
Common misconception
Not every data set is normal. Income and reaction times are strongly skewed, so the 68-95-99.7 rule does not apply to them.
INVESTIGATION
Try it yourself
Collect 40 measurements, such as the time to complete a short task. Calculate the mean and standard deviation, count how many values fall within one standard deviation and compare with the expected 68%.
Watch
Basic Probability
Math Antics · 11:28
A refresher on probability language before the formal rules.
Before you watch: Are drawing two cards with replacement independent events?
- Compute P(A or B) for mutually exclusive events with P(A)=0.3, P(B)=0.45.
- How many ways can 3 students be chosen from 10?
Interactive simulation · PhET
Plinko Probability
Run 5000 trials and compare the resulting shape to a normal curve.
While you explore
- Roughly what proportion of results falls within one standard deviation of the mean?
- Why do so many random processes produce a bell shape?
Key vocabulary
- Standard deviation
- A measure of how spread out data is around the mean.
- Combination
- A selection where the order does not matter.
Practice questions
0/1 correct
Level 1 · Criterion A
Two fair dice are rolled. What is P(both show a six)?
MYP criterion tasks
Level 2 · Criterion D
A committee of 3 is chosen from 12 people. How many committees are possible, and why is this a combination?
Reflect & track
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