Theoretical and experimental probability
Systematically listing outcomes for a sample space, finding theoretical probability, and comparing it to experimental probability from real trial data
About three lessons of 45 to 60 minutes
36 outcomes, hiding in plain sight
Roll two dice, and there are exactly 36 equally likely outcomes, from (1,1) up to (6,6), even though the SUMS they produce range unevenly from 2 to 12. Most people guess every sum is equally likely, but a sum of 7 has 6 different ways to happen (1+6, 2+5, 3+4, 4+3, 5+2, 6+1), while a sum of 2 has only 1 (1+1). Systematically listing every outcome, rather than guessing, is the only reliable way to find a theoretical probability.
Theoretical probability predicts what SHOULD happen, based on counting equally likely outcomes. Experimental probability describes what actually DID happen in real trials, and the two do not always match exactly, especially with fewer trials. This unit builds both skills, and the judgement to compare them sensibly.
- Two dice, sum = 7the most likely sum, with 6 of the 36 equally likely outcomes
- A coin and a die togetherhave 2 x 6 = 12 equally likely outcomes
- A die rolled 40 times, a 3 comes up 5 timesexperimental probability 5/40 = 1/8, lower than the theoretical 1/6
- Three coin flips8 equally likely outcomes, from HHH to TTT
What students will be able to do
Students will systematically list every outcome in a sample space for combined events, calculate theoretical probability as a fraction from that sample space, use the counting principle to find the total number of outcomes for multi-stage choices, and find the experimental probability from real trial data and compare it to the theoretical probability, explaining any difference.
- I can systematically list every outcome for a combined event, so none are missed.
- I can find the theoretical probability of an event as a fraction, from a sample space.
- I can use the counting principle to find the total number of outcomes for a multi-stage choice.
- I can find the experimental probability from trial data, as a fraction or decimal.
- I can compare an experimental probability to the theoretical probability, and explain whether it is higher, lower, or about the same.
Standards this unit teaches
- Phase 4 (Years 9-10): Probability, Experimental and theoretical probabilityNew Zealand Curriculum (NZC), Mathematics and StatisticsExperimental and theoretical probability
Paraphrased (see the licence note in lib/content_nzsecondarymath2.ts) from the Ministry of Education's Tāhūrangi curriculum site, "NZC - Mathematics and Statistics Phase 4 (Years 9-10)" (official policy for all English-medium state and state-integrated schools from 1 January 2026), Probability strand, "Experimental and theoretical probability" section: during Year 9, students systematically list every outcome in a sample space (using lists, tables, two-way tables or tree diagrams), calculate theoretical probability from that sample space, and compare it with the experimental probability found from at least 30 trials of a chance experiment, explaining why the two can differ and how increasing the number of trials tends to narrow the gap. (Year 10 raises the trial count to 100+ to demonstrate the Law of Large Numbers directly, and is deliberately left out of this unit.) Source: https://newzealandcurriculum.tahurangi.education.govt.nz/nzc---mathematics-and-statistics-phase-4-years-9-10/5637291579.p (verified live 2026-07-14).
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
Words to teach and display
- Sample space
- the full list of every possible outcome of a chance experiment
- Theoretical probability
- the probability predicted by counting equally likely outcomes, before any trials are actually run
- Experimental probability
- the probability calculated from real trial data: number of times an event happened, divided by the total number of trials
- Counting principle
- multiplying the number of choices at each stage to find the total number of possible combinations
- Tree diagram
- a branching diagram that systematically shows every outcome of a multi-stage chance experiment
- Relative frequency
- another name for experimental probability: how often an event occurred relative to the total number of trials
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. Theoretical probability from a sample space
ConcreteTwo fair dice have 6 x 6 = 36 equally likely outcomes altogether, from (1,1) to (6,6). Theoretical probability is found by counting how many of those 36 outcomes satisfy the event, then writing that count over 36 (simplified if possible).
For 'the sum is 4', the outcomes are (1,3), (2,2) and (3,1): exactly 3 out of 36, so the theoretical probability is 3/36, which simplifies to 1/12. Listing outcomes systematically (in order, without skipping any) is the only reliable way to count correctly, guessing at the count almost always gets it wrong.
The same counting idea applies to any pair of independent events, not just two dice: a coin and a 6-sided die together have 2 x 6 = 12 equally likely outcomes, and a spinner with n equal sectors has n equally likely outcomes, one per sector.
Two fair dice are rolled. List the outcomes where the sum is 4, and find the theoretical probability.
- List the outcomes: (1,3), (2,2), (3,1). That is 3 outcomes.
- There are 36 equally likely outcomes in total (6 x 6).
- Probability = 3/36 = 1/12.
Answer: The probability the sum is 4 is 3/36, which simplifies to 1/12.
- Why are there 6 different ways to roll a sum of 7 with two dice, but only 1 way to roll a sum of 2?
- How many total equally likely outcomes are there for a coin flipped alongside a 6-sided die rolled?
2. Systematic listing: tables, tree diagrams and the counting principle
PictorialListing outcomes systematically means working through every combination in a fixed order, so none get missed and none get repeated. For a coin (heads or tails) and a 4-sided die (1 to 4), listing heads first with every die value, then tails with every die value, gives all 8 outcomes without gaps.
The counting principle finds the TOTAL number of outcomes without listing every one individually: multiply the number of choices at each separate stage. Choosing one of 3 drink sizes, one of 4 flavours and one of 2 toppings gives 3 x 4 x 2 = 24 total combinations, without writing all 24 out.
A tree diagram extends the same systematic idea across more than two stages. Three coin flips branch into 2 x 2 x 2 = 8 outcomes (HHH, HHT, HTH, HTT, THH, THT, TTH, TTT), and counting how many branches match a condition (like 'exactly 2 heads') gives its probability directly.
A coin is flipped and a 4-sided die (1 to 4) is rolled. List all outcomes, and find the probability of heads and a number greater than 2.
- List all 8 outcomes: (H,1), (H,2), (H,3), (H,4), (T,1), (T,2), (T,3), (T,4).
- Favourable outcomes (heads AND greater than 2): (H,3), (H,4). That is 2 outcomes.
- Probability = 2/8 = 1/4.
Answer: There are 8 total outcomes, and the probability of heads and a number greater than 2 is 2/8, which simplifies to 1/4.
- Why does listing 'heads with every die value, then tails with every die value' guarantee no outcome is missed or repeated?
- If a menu has 3 starters, 5 mains and 2 desserts, how many total meal combinations are possible, without listing them all?
3. Comparing experimental and theoretical probability
AbstractExperimental probability (also called relative frequency) is calculated from real trial data: the number of times an event happened, divided by the total number of trials. It is not always exactly equal to the theoretical probability, especially with a smaller number of trials.
A die rolled 40 times that shows a 3 exactly 5 times has experimental probability 5/40 = 1/8 = 0.125. The theoretical probability of rolling a 3 is 1/6, about 0.167. Since 0.125 is less than 0.167, the experimental probability is LOWER than the theoretical probability, in this particular set of trials.
This difference is expected, not a mistake: with a limited number of trials, chance variation means the experimental result will rarely land exactly on the theoretical value. As the number of trials increases, the experimental probability tends to get closer to the theoretical probability (this idea, the Law of Large Numbers, is explored further with 100+ trials in Year 10).
A die is rolled 40 times, and a 3 comes up 5 times. Find the experimental probability, and compare it to the theoretical probability of 1/6.
- Experimental probability = 5/40 = 1/8 = 0.125.
- Theoretical probability = 1/6, about 0.167.
- 0.125 is less than 0.167, so the experimental probability is lower than the theoretical probability.
Answer: The experimental probability is 1/8 (0.125), which is lower than the theoretical probability of 1/6 (about 0.167).
- Is it a mistake if an experimental probability does not exactly equal the theoretical probability? Why or why not?
- What tends to happen to the gap between experimental and theoretical probability as the number of trials increases?
Common misconceptions and how to address them
MisconceptionWith two dice, every possible sum (2 through 12) is equally likely, because there are 11 possible sums.
Why it happens: Students count the number of DIFFERENT sums rather than the number of OUTCOMES that produce each sum, not realising some sums (like 7) can be made in more ways than others (like 2 or 12).
How to address it: List the actual outcomes for two or three different sums side by side: sum = 2 has only (1,1), one outcome, while sum = 7 has (1,6), (2,5), (3,4), (4,3), (5,2), (6,1), six outcomes. More ways to make a sum means a higher probability for that sum.
MisconceptionIf an experimental probability does not exactly match the theoretical probability, an error was made somewhere.
Why it happens: Students expect real trial data to reproduce the theoretical prediction exactly, not accounting for natural chance variation, especially over a smaller number of trials.
How to address it: Explain that experimental probability is EXPECTED to vary from the theoretical value, particularly with fewer trials; the gap tends to narrow as more trials are run, but rarely closes to exactly zero.
MisconceptionThe counting principle means ADDING the number of choices at each stage, not multiplying.
Why it happens: Students confuse the counting principle (used for combining independent choices, one from each of several categories) with simple addition (used for combining separate, non-overlapping options within ONE category).
How to address it: Contrast the two directly: choosing ONE drink from 5 options is 5 choices (no multiplication needed). Choosing one drink from 5 AND one snack from 3 is 5 x 3 = 15 combinations, because each of the 5 drink choices can be paired with each of the 3 snack choices.
Guided practice (with answers)
1. Two fair dice are rolled. Find the probability that both show an odd number.
Answer: 1/4, because each die has 3 odd values (1, 3, 5), giving 3 x 3 = 9 outcomes out of 36: 9/36 = 1/4.
2. A coin is flipped and a 6-sided die is rolled. Find the probability of tails and an even number.
Answer: 1/4, because tails pairs with 3 even values (2, 4, 6) out of the 12 total outcomes: 3/12 = 1/4.
3. List all the possible outcomes when flipping two fair coins.
Answer: (heads, heads), (heads, tails), (tails, heads), (tails, tails), 4 outcomes in total.
4. A shop offers 3 drink sizes and 5 flavours. How many different combinations are possible?
Answer: 15, because 3 x 5 = 15 (the counting principle).
5. A die is rolled 50 times and shows a 2 on 6 of them. Find the experimental probability as a fraction.
Answer: 3/25, because 6/50 simplifies to 3/25.
6. Is 3/25 higher or lower than the theoretical probability of 1/6 for rolling a 2?
Answer: Lower: 3/25 = 0.12, which is less than 1/6, about 0.167.
Independent practice worksheets
Practise theoretical probability, systematic listing, and comparing experimental to theoretical probability, with computed, never-wrong answer keys.
Differentiation
- For two-dice probability, provide a printed 6-by-6 grid to fill in every sum before counting, rather than expecting students to enumerate outcomes mentally.
- For systematic listing, insist on one fixed order every time (e.g. always list the coin result first, then every die value for that coin result), so no outcome gets missed or duplicated.
- For the counting principle, write the multiplication out explicitly as 'choices x choices x choices' before calculating, rather than jumping straight to the final number.
- For experimental vs theoretical comparisons, always calculate BOTH values as decimals first, then compare the two decimals directly, rather than trying to compare fractions with different denominators by eye.
- Investigate: for two dice, which sum has the highest theoretical probability, and why (hint: count the outcomes for every sum from 2 to 12 and look for a pattern)?
- Design and carry out a real chance experiment (e.g. rolling a die or flipping a coin, at least 30 times), record the results, and compare the experimental probability to the theoretical prediction.
- Extend the counting principle to a 4-stage choice (e.g. size, flavour, topping and packaging) and calculate the total number of combinations.
- Preview the Law of Large Numbers (next year's Year 10 topic) by predicting: if the same die-rolling experiment were repeated with 100 trials instead of 40, would you expect the experimental probability to be closer to, or further from, the theoretical value? Why?
Assessment: exit ticket
A three-question exit ticket sampling two-dice theoretical probability, a 3-coin tree diagram, and an experimental-vs-theoretical comparison.
1. Two fair dice are rolled. Find the probability that the sum is 8.
Answer: 5/36, because the outcomes (2,6), (3,5), (4,4), (5,3), (6,2) give 5 out of 36.
2. Three coins are flipped. Find the probability of getting exactly 1 head.
Answer: 3/8, because HTT, THT and TTH are the 3 outcomes (of 8 total) with exactly 1 head.
3. A coin is flipped 60 times and lands on heads 36 times. Find the experimental probability as a decimal, and state whether it is higher than, lower than, or equal to the theoretical probability of 0.5.
Answer: 0.6, which is higher than 0.5, because 36/60 = 0.6.
Teacher notes and timings
- Rough timing across three lessons: Lesson 1 theoretical probability from a sample space (section 1), Lesson 2 systematic listing, tree diagrams and the counting principle (section 2), Lesson 3 comparing experimental to theoretical probability, plus the exit ticket (section 3).
- This is the site's fourth teaching unit anchored to the New Zealand Curriculum (NZC). See the licence note at the top of lib/content_nzsecondarymath2.ts: the NZC's Tāhūrangi content is licensed CC BY-NC 4.0 (NonCommercial), so this unit paraphrases the curriculum rather than quoting it, with the source URL cited for attribution.
- Scope note: this unit deliberately covers Year 9's probability practices (systematic listing, theoretical probability, and comparing to experimental probability from at least 30 trials) and leaves Year 10's continuation (100+ trials to directly demonstrate the Law of Large Numbers) for a later unit, matching how the source page itself splits Year 9 from Year 10.
- Language to keep repeating: theoretical probability predicts what SHOULD happen from counting, experimental probability describes what DID happen from real trials, and a gap between the two is expected, not an error.
- Use Student view to project this lesson. Print saves the full teacher unit, including answers and teacher notes; use the linked independent-practice worksheets for student handouts.