Probability Foundations

Overview

Students begin a four-week probability unit by interpreting likelihood in familiar contexts and developing the language needed for probability problems. They move from the probability scale to calculating and communicating simple theoretical probabilities, building foundations for later work with compound events, tables, tree diagrams and distributions.

Learning intentions

• WALT define and use the terms outcome, event, sample space and favourable outcome.
• WALT calculate simple theoretical probabilities from equally likely outcomes.
• WALT express probabilities as fractions, decimals and percentages.
• WALT explain what a probability tells us in a real-world context.

Success criteria

• I can place an event on a probability scale from 0 to 1.
• I can identify the sample space and favourable outcomes.
• I can use (P(E)=\frac{\text{number of favourable outcomes{\text{total number of equally likely outcomes).
• I can check that my answer is between 0 and 1 and communicate it accurately.

Curriculum links

• Interpret and apply mathematical and statistical information in context by making informed judgements from probability statements and representations.
• Demonstrate mathematical reasoning by using appropriate probability methods, notation and representations.
• Use mathematical methods to explore problems relating to life in Aotearoa New Zealand or the Pacific through accurate calculations and contextual explanations.
• Mathematical and statistical thinking: connect representations, justify conclusions and communicate reasoning clearly.

Lesson structure (90 minutes)

1. 0–10 min · Probability hook. Teacher opens with the probability hook and learning objectives and displays four statements: “The sun will rise tomorrow”, “You will roll a 7 on a standard die”, “A fair coin will land heads”, and “You will draw a red card from a standard deck”; students rank them from impossible to certain, then justify one choice with a partner. Teacher elicits that probability describes chance and introduces the scale from 0 to 1, including the language impossible, unlikely, even chance, likely and certain.

2. 10–35 min · Direct instruction and modelling. Teacher uses the vocabulary and worked-example slides to define outcome, event, sample space and favourable outcome, then models (P(E)=\frac{\text{favourable outcomes{\text{total outcomes) using a die, a deck of cards and a bag of coloured counters. Students annotate the probability definitions and examples worksheet, convert selected fractions to decimals and percentages, and complete quick checks such as (P(\text{rolling a 4})=\frac16) and (P(\text{drawing a heart})=\frac14). Teacher emphasises that probabilities must lie from 0 to 1 inclusive.

3. 35–60 min · Guided practice. Teacher works through the guided-practice questions with the class, asking students to identify the sample space before calculating: rolling an even number, selecting a vowel from “PROBABILITY”, choosing a marble from a bag containing 5 red, 3 blue and 2 green, and selecting a girl from a group of 18 girls and 12 boys. Students solve independently for two minutes, compare methods in pairs, and share answers using complete statements such as “The probability is … because …”. Teacher checks whether students are counting total and favourable outcomes correctly and addresses misconceptions immediately.

4. 60–82 min · Independent application. Teacher distributes the independent probability practice questions and uses the independent-practice instructions to set expectations: show working, simplify fractions where possible, and give a decimal or percentage when requested. Students complete a selection of questions involving a die, an eight-section spinner, coloured balls and a standard deck of cards. Early finishers investigate the LOTTO context: calculate the probability of drawing an even number, a number less than 5, and a number less than 26 from 40 equally likely balls, then explain which event is more likely and why.

5. 82–90 min · Plenary and exit check. Teacher returns to the plenary and exit-ticket slide and asks students to explain why a probability of (\frac{7}{5}) cannot be valid. Students complete the worksheet exit ticket: “A bag has 4 black, 3 white and 5 blue counters. Find the probability of selecting white, express it as a fraction and decimal, and explain what the answer means.” Students hand this in as they leave.

Resources

• the complete probability lesson deck
• the probability definitions, practice questions and exit ticket
• Standard six-sided die
• Fair coin
• Standard pack of playing cards
• Bag or container with coloured counters
• Board and markers
• Calculators, optional for checking decimal conversions

Assessment

• Listen to partner explanations during the ranking task and question students about their use of probability vocabulary.
• Check guided-practice responses for correct identification of the sample space, favourable outcomes, fraction simplification and contextual statements.
• Use the exit ticket to identify students needing support with total outcomes, fraction-to-decimal conversion or interpreting probability.

Differentiation

• Provide a dyslexia-friendly version of the probability worksheet using a clear sans-serif font, generous spacing, uncluttered page design and key vocabulary with definitions. Read questions aloud and allow students to use text-to-speech or coloured overlays where helpful.
• Support learners with a visible probability scale, a worked example, counters for physically representing outcomes, and sentence starters: “The sample space is…”, “The favourable outcomes are…”, and “Therefore, the probability means…”.
• Pair students strategically for guided practice and reduce the number of independent questions while retaining the key question types. Allow calculators after students have set up the correct fraction.
• Extend advanced learners with the LOTTO comparisons, requiring a general explanation of how changing the number of favourable outcomes affects probability and a justified ranking of several events.