Estimating Experimental Probability

Overview

This is lesson 4 of 8 in the Year 10 unit Area and Probability. Students use repeated trials to estimate probability, compare experimental and theoretical values, and investigate how sample size affects reliability. The lesson also connects probability with estimating area.

Learning intentions

Students will:

• Estimate probability from repeated trials using relative frequency.
• Compare experimental and theoretical probability.
• Explain variation in results and the effect of increasing sample size.
• Record, combine and represent data using tables and graphs.
• Connect experimental proportions with area fractions.

Success criteria

• I can record trial results systematically.
• I can calculate relative frequency as successes divided by trials.
• I can compare experimental and theoretical probability using appropriate language.
• I can explain why a small number of trials may produce an unreliable estimate.

Curriculum links

• Represent and interpret probability using fractions, decimals and percentages.
• Conduct probability experiments and use data to estimate likelihood.
• Compare experimental probability with theoretical probability and explain variation.
• Represent data in tables and graphs, and communicate mathematical reasoning.
• Apply proportional reasoning to estimate area from a randomly selected sample.

Lesson structure (60 minutes)

1. 0–5 minutes – Hook and retrieval

Open with the introduction and hook slides. Ask: “If a coin lands heads five times in a row, is tails now more likely?” Students give a quick individual response, then discuss with a partner. Review the meaning of probability and distinguish a probability value from a number of successes.

2. 5–15 minutes – Explicit teaching and worked example

Use the teaching and worked-example slides to introduce experimental probability:

[ P(A)\approx\frac{\text{number of successes{\text{number of trials ]

Model a relative frequency table and the example: in 40 die rolls, a six occurs 9 times, so the experimental probability is (9/40=0.225=22.5%), compared with the theoretical probability (1/6\approx16.7%). Emphasise that this difference is plausible because of random variation. Explain that larger samples generally produce more stable estimates, but do not guarantee an exact result.

3. 15–18 minutes – Checkpoint

Students complete the checkpoint on the experimental probability investigation worksheet: identify which statement is a probability and which is a number of successes, then calculate a relative frequency. Quickly scan responses and address errors before the investigation.

4. 18–35 minutes – Coin-toss investigation

Place students in groups of four, assigning roles of tosser, recorder, checker and equipment manager. Each group conducts 100 coin tosses, physically or using a simulation, and records cumulative numbers of heads and relative frequencies after 10, 25, 50 and 100 trials on the recording table and graph section. Remind students to use the same success outcome, record every trial and avoid resetting results.

5. 35–43 minutes – Combine and graph class results

Groups contribute their cumulative data to a class table. Use the data display and graphing slides to guide students in graphing relative frequency against number of trials, either by hand or in a spreadsheet. Discuss whether the class graph tends to become more stable as the number of trials increases and why different groups may have different results.

6. 43–53 minutes – Area connection

Give each group a 10 by 10 grid and ask them to mark or identify a target region, then randomly select squares to estimate the shaded area fraction. Students compare the experimental proportion of selected shaded squares with the exact area ratio of the shaded region. Record the comparison and explanation on the area connection questions. Discuss how random selection provides an estimate of area, while counting the grid can provide the exact ratio.

7. 53–60 minutes – Plenary and exit response

Use the discussion and plenary slides for a final class discussion. Students complete the exit response: “Why are 10 trials generally less reliable than 1,000 trials?” They must use the terms sample size, variation and experimental probability. Set the homework: write a short comparison of the theoretical and experimental results from the investigation, including one possible source of variation.

Resources

• the complete lesson slide deck
• the experimental probability investigation worksheet
• Coins or a probability simulation
• Dice for the worked example or checking
• 10 by 10 grids and coloured pencils
• Spreadsheet software or graph paper
• Board and markers
• Calculators

Assessment

• Checkpoint responses: distinguish probability from number of successes and calculate relative frequency.
• Observe group recording, role participation, accuracy and graph interpretation during the investigation.
• Exit response and homework: explain the effect of sample size and compare experimental with theoretical probability.

Differentiation

• Provide a recording table with headings, worked first-row examples, a formula box and a partially completed graph.
• Support students who need assistance by pairing them with a confident recorder and providing sentence starters such as “The experimental probability is higher/lower because…”.
• Allow students with numeracy, fine-motor or processing needs to use a simulation, calculator, enlarged grid or shared group recording role.
• Extend students by investigating how many trials are needed before the estimate appears stable, comparing groups’ results, and discussing possible bias such as an uneven coin or inconsistent recording.