Probability models and experimental probability
Understanding probability as a number from 0 to 1, building probability models, and comparing them to observed results
About three lessons of 45 to 60 minutes
A coin lands heads 8 times in a row. Is it 'due' for tails?
No: each flip is independent, and the coin has no memory of the last one. But a real spinner, tested 200 times, almost never lands EXACTLY on its theoretical prediction either. Both of these ideas, that probability is a fixed number describing long-run behaviour, and that real experiments only approach that number rather than hitting it exactly, are the heart of this unit.
Every probability, from a die roll to a weather forecast, is a single number between 0 (impossible) and 1 (certain). Building a probability MODEL means working out that number in theory; running an EXPERIMENT and counting results tells you what actually happened, and comparing the two is how you check whether a model is trustworthy.
- Rolling a die: theoretical probability of a 6 is 1/6the model, worked out before any dice are rolled
- A spinner tested 200 times, landing on red 61 timesthe experimental (observed) probability, found by counting
- A weather forecast: '30% chance of rain'a probability model built from historical weather data
- A coin that has landed heads 8 times in a roweach flip is independent; the coin is never 'due' for tails
What students will be able to do
Students will understand probability as a number from 0 to 1 that describes how likely an event is, build a theoretical probability model for a simple event, and compare theoretical (model) probabilities to experimental (observed) probabilities from real or simulated trials.
- I can place a probability on a scale from 0 (impossible) to 1 (certain).
- I can find the theoretical probability of an event as a fraction, favourable outcomes over total outcomes.
- I can find the experimental (observed) probability of an event from the results of a set of trials.
- I can compare a theoretical probability to an experimental one and explain why they are often close but rarely identical.
Standards this unit teaches
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
Words to teach and display
- Probability
- a number from 0 to 1 describing how likely an event is to happen
- Sample space
- the complete list of all possible outcomes of an experiment
- Theoretical probability
- a probability calculated in advance, favourable outcomes divided by total outcomes, assuming every outcome is equally likely
- Experimental (observed) probability
- a probability found by actually running trials and counting how often an event happened, out of the total number of trials
- Relative frequency
- another name for experimental probability, the fraction of trials in which an event occurred
Teach it: concrete, pictorial, abstract
The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.
1. Probability as a number from 0 to 1
ConcreteEvery probability sits somewhere on a single scale from 0 to 1: 0 means the event is impossible, 1 means it is certain, and 1/2 (or 0.5) means an even chance. The closer to 1, the more likely; the closer to 0, the less likely.
A bag has 3 red marbles and 7 blue marbles (10 total). The probability of drawing red is 3/10 = 0.3, which sits between impossible (0) and even chance (0.5), closer to unlikely than to a coin flip.
A bag has 3 red marbles and 7 blue marbles. Find the probability of drawing a red marble.
- Favourable outcomes (red marbles): 3.
- Total outcomes (all marbles): 10.
- Probability = 3/10 = 0.3.
Answer: P(red) = 3/10, or 0.3.
- What does a probability of exactly 0 mean about an event?
- Is P(red) = 0.3 closer to impossible or to an even chance?
2. Building a probability model
PictorialA probability model lists every outcome in the sample space with its theoretical probability, calculated before any trial is run, by assuming every outcome is equally likely.
A spinner has 4 equal sections: red, blue, green, yellow. Since the sections are equal, each has theoretical probability 1/4 = 0.25. The probability of NOT landing on red is 1 - 1/4 = 3/4, because 'not red' and 'red' together must account for the whole sample space (they sum to 1).
A spinner has 4 equal sections: red, blue, green, yellow. Find P(red) and P(not red).
- Each of the 4 equal sections has probability 1/4.
- P(red) = 1/4.
- P(not red) = 1 - 1/4 = 3/4.
Answer: P(red) = 1/4; P(not red) = 3/4.
- Why must P(red) and P(not red) always add up to exactly 1?
- If a spinner had 5 equal sections instead of 4, what would the probability of any one section be?
3. Comparing experimental results to the model
AbstractRunning real trials and counting results gives the EXPERIMENTAL probability, which is expected to land close to the theoretical model but rarely matches it exactly, especially with a smaller number of trials. As the number of trials grows, experimental probability tends to get closer to the theoretical value.
The same 4-section spinner (theoretical P(red) = 1/4 = 0.25) is spun 200 times, and lands on red 61 times. Experimental probability: 61/200 = 0.305. This is close to the theoretical 0.25 but not identical, which is completely expected: it reflects natural variation, not a broken spinner.
A spinner has 4 equal sections, so each has theoretical probability 1/4. In 200 spins, red came up 61 times. Find the experimental probability of red, and compare it with the theoretical probability.
- Experimental probability = observed count / total trials = 61 / 200.
- 61 / 200 = 0.305.
- Compare: theoretical is 0.25 (1/4), experimental is 0.305. They are close but not equal.
Answer: Experimental probability of red = 0.305, compared to a theoretical probability of 0.25. The two are close, which is expected, since a real experiment only approaches the theoretical model, and would be expected to get even closer with more trials.
- Why would you expect the experimental probability to get closer to 0.25 if the spinner were spun 2000 times instead of 200?
- Does a difference between the experimental and theoretical probability always mean the spinner is unfair? Why or why not?
Common misconceptions and how to address them
MisconceptionA probability of 0.5 means an event will happen exactly half the time in ANY small number of trials, such as exactly 5 heads in 10 flips.
Why it happens: Students treat a long-run theoretical probability as a guarantee for a small, specific sample.
How to address it: A probability of 0.5 describes the LONG-RUN tendency over many, many trials. A small number of trials, like 10 coin flips, can easily deviate from that, showing 3, 4, 6 or 7 heads without anything being wrong.
MisconceptionProbability can be found by counting favourable outcomes alone, without dividing by the total number of outcomes.
Why it happens: Students report the count (e.g. '3 red marbles') as if it were already the probability, skipping the division.
How to address it: Probability is always favourable outcomes DIVIDED BY total outcomes. A probability is always a number between 0 and 1, never a raw count.
MisconceptionAfter several heads in a row, a coin is 'due' for tails, because the results should even out.
Why it happens: This is the well-documented gambler's fallacy: assuming past independent outcomes affect the next one.
How to address it: Each flip is an independent event: the coin has no memory. The probability of heads on the NEXT flip is still exactly 0.5, no matter what happened on previous flips.
MisconceptionA probability of 1/4 means the event will happen exactly 1 time out of the very next 4 tries.
Why it happens: Students read a probability as a short-term guarantee rather than a long-run proportion.
How to address it: A probability of 1/4 describes what happens on AVERAGE over many trials, not a promise for any specific small group of tries. It might happen 0 times, or 2 times, in any particular set of 4 tries.
MisconceptionAny difference between an experimental result and the theoretical probability means the model or the object (die, spinner, coin) must be unfair.
Why it happens: Students expect an exact match and treat any gap as evidence of a problem.
How to address it: Some variation between experimental and theoretical probability is completely normal, especially with fewer trials. Only a LARGE or persistent difference across MANY trials is good evidence that a model might be wrong.
Guided practice (with answers)
1. A bag has 5 green and 15 yellow marbles (20 total). Find P(green) as a fraction and decimal.
Answer: 1/4, or 0.25, because 5/20 simplifies to 1/4.
2. A number cube (1 to 6) is rolled. Find P(rolling an even number).
Answer: 1/2, or 0.5, because 3 of the 6 outcomes (2, 4, 6) are even, and 3/6 = 1/2.
3. A spinner has 8 equal sections numbered 1 to 8. Find P(not landing on 8).
Answer: 7/8, or 0.875, because 7 of the 8 sections are not 8.
4. In 50 trials of flipping a coin, heads came up 28 times. Find the experimental probability of heads.
Answer: 0.56, because 28/50 = 0.56.
5. The theoretical probability of a coin landing heads is 0.5. In 400 flips, about how many heads would you expect?
Answer: About 200, because 0.5 x 400 = 200.
6. A die is rolled 60 times and lands on 6 fifteen times. Find the experimental probability of rolling a 6, and compare it to the theoretical probability of 1/6.
Answer: Experimental probability = 15/60 = 0.25, noticeably higher than the theoretical 1/6 (about 0.167) for this particular sample of 60 rolls.
Independent practice worksheets
Practise probability, sample space and experimental probability with computed, never-wrong answer keys.
Differentiation
- Keep the probability scale visible throughout the unit, and have students physically place each new probability on it before moving on.
- Practice writing the favourable-over-total fraction as its own explicit first step before simplifying or converting to a decimal.
- Use very small numbers of trials (10 or 20) for the first experimental-probability examples, then move to the 200-trial spinner example.
- Explicitly separate 'what the model predicts' from 'what actually happened' with two labeled columns on the board for every comparison problem.
- Simulate 100 coin flips (physically or with an online simulator) and compare the class's actual experimental probability to the theoretical 0.5.
- Introduce a probability model for a NON-equally-likely event, such as a weighted spinner, and discuss how the model's probabilities would differ from the equal-sections case.
- Ask students to predict how much closer to the theoretical value they would expect the experimental probability to get if the number of trials were multiplied by 10.
- Explore compound scenarios: what is the theoretical probability of landing on red twice in a row on the 4-section spinner? (1/4 x 1/4 = 1/16, previewing later compound-probability work.)
Assessment: exit ticket
A three-question exit ticket sampling a theoretical probability, the meaning of a probability, and an experimental-vs-theoretical comparison.
1. A jar has 4 red, 6 blue and 10 white marbles (20 total). Find P(blue).
Answer: 3/10, or 0.3, because 6/20 simplifies to 3/10.
2. A probability of an event is 0. What does this mean?
Answer: The event is impossible; it cannot happen.
3. A spinner with 4 equal sections is spun 120 times and lands on red 42 times. Find the experimental probability of red and compare it to the theoretical probability of 1/4.
Answer: Experimental probability = 42/120 = 0.35, somewhat higher than the theoretical 0.25 (1/4). This is within the range of normal sampling variation for 120 trials, not necessarily evidence the spinner is unfair.
Teacher notes and timings
- Rough timing across three lessons: Lesson 1 the 0-to-1 probability scale (section 1), Lesson 2 building a probability model (section 2), Lesson 3 comparing experimental to theoretical probability plus the exit ticket (section 3 and assessment).
- This unit assumes comfort with simplifying fractions (Grade 6 dividing fractions unit touches this skill). Revisit fraction simplification first if reducing a probability like 6/20 to 3/10 is shaky.
- Language to keep repeating: probability is always favourable over total, between 0 and 1; theoretical is the model calculated in advance, experimental is what trials actually showed; more trials bring experimental closer to theoretical, but rarely make them identical.
- The bar-model comparison figure in section 3 deliberately shows the EXPECTED count (from the model) next to the OBSERVED count (from the experiment) on the same scale, so the gap between them is visible rather than just two separate numbers.
- Curriculum note: 7.SP.C.5 (the 0-to-1 scale) and 7.SP.C.7 (building and testing models) are grouped as one unit here because 7.SP.C.7's model-building is not meaningful without first understanding what the probability number itself represents.
- Present mode and print both work: use Present to build the probability-scale number line live with the class in section 1, then print the worksheets for independent practice.