Review of Basic Probability

This topic asks students to shift from calculating basic probabilities to reasoning about how information updates our expectations. Active simulations and collaborative tasks let them experience firsthand how 'given that' conditions reshape likelihood, making abstract ideas concrete and memorable.

• Year 11
• Mathematics
• 3 activities
• 25 min–50 min

Learning Objectives

1. Construct a sample space for at least three different random experiments.
2. Differentiate between theoretical and experimental probability using examples.
3. Calculate the probability of simple and compound events.
4. Analyze how the size of the sample space impacts the probability of an event occurring.
5. Compare theoretical and experimental probabilities derived from simulations.

Simulation Game

⏱ 40 min·Pairs

Simulation Game: The Monty Hall Challenge

Students run a simulation of the famous 'three doors' game in pairs. They record the results of 'staying' vs 'switching' over 20 rounds, then use conditional probability formulas to explain why switching doubles their chances of winning.

Prepare & details

Differentiate between theoretical and experimental probability.

Facilitation Tip: During The Monty Hall Challenge, circulate and listen for students who use phrases like 'it feels like switching' instead of computing probabilities; this is your signal to prompt them to calculate the exact chance of winning if they stay versus switch.

Setup: Flexible space for group stations

Materials: Role cards with goals/resources, Game currency or tokens, Round tracker

• Apply
• Analyze
• Evaluate
• Create
• Social Awareness
• Decision-Making

Inquiry Circle

⏱ 50 min·Small Groups

Inquiry Circle: Medical Testing Logic

Groups are given data about a hypothetical disease and a diagnostic test with a known error rate. They must use a tree diagram to find the probability that a person actually has the disease *given* they tested positive, discussing the implications for public health.

Prepare & details

Analyze how the size of the sample space affects the probability of an event.

Facilitation Tip: For Medical Testing Logic, provide red and blue counters so students can physically model sensitivity and false positive rates before moving to abstract notation.

Setup: Groups at tables with access to source materials

Materials: Source material collection, Inquiry cycle worksheet, Question generation protocol, Findings presentation template

• Analyze
• Evaluate
• Create
• Self-Management
• Self-Awareness

Think-Pair-Share

⏱ 25 min·Pairs

Think-Pair-Share: Are They Independent?

Students are given pairs of real-world events (e.g., 'wearing a hat' and 'getting a sunburn'). They must individually decide if they are independent, then pair up to prove it using the formula P(A∩B) = P(A)P(B).

Prepare & details

Construct a sample space for a given random experiment.

Facilitation Tip: In Are They Independent?, give each pair a different pair of events written on sticky notes so you can rotate and listen to multiple conversations rather than hearing the same pair repeatedly.

Setup: Standard classroom seating; students turn to a neighbor

Materials: Discussion prompt (projected or printed), Optional: recording sheet for pairs

• Understand
• Apply
• Analyze
• Self-Awareness
• Relationship Skills

Teaching This Topic

Teachers often begin with concrete experiments so students feel the tension between intuition and calculation. Avoid rushing to formulas; instead, let students derive conditional probability from frequency tables they build themselves. Research shows that drawing tree diagrams by hand, even messily, helps students map dependencies accurately and reduces later errors with notation.

What to Expect

By the end of these activities, students should confidently distinguish independent events from mutually exclusive ones, use formal notation correctly, and justify their conclusions with diagrams and data rather than intuition alone.

Watch Out for These Misconceptions

Common Misconception

During Are They Independent?, watch for students who label events as independent simply because they seem unrelated in everyday language.

What to Teach Instead

Have them calculate P(A), P(B), and P(A and B) from their Venn diagram and check if P(A and B) equals P(A) × P(B); if not, the events are dependent regardless of intuition.

Common Misconception

During The Monty Hall Challenge, listen for students who claim the remaining unopened door has a 50% chance after one goat is revealed.

What to Teach Instead

Ask them to list all possible initial choices and host actions, then compute the exact win probability for switching versus staying using the data they collect.

Assessment Ideas

Quick Check

After The Monty Hall Challenge, ask students to write the conditional probability of winning if they stay with their first choice given that Monty opens Door 3, using both notation and a brief explanation of their calculation.

Discussion Prompt

During Medical Testing Logic, after groups present their results, pose this prompt: 'If a test is 95% accurate, why might a positive result still be unlikely to indicate the disease in a population where only 1% have it?' Circulate and listen for references to false positives and base rates.

Exit Ticket

After Are They Independent?, give each student a mini whiteboard with two events written on it (e.g., 'rolling a 4 on a die' and 'flipping heads on a coin'). Ask them to draw a Venn diagram and state whether the events are independent, justifying with both a diagram and a calculation.

Extensions & Scaffolding

• Challenge: Ask students to design their own conditional probability scenario using real data they find online and present it to the class.
• Scaffolding: Provide partially completed Venn or tree diagrams with missing labels for students who need structure to see the connections.
• Deeper exploration: Introduce Bayes’ Theorem through a guided worksheet that connects the Medical Testing Logic results back to the formula P(A|B) = P(B|A) × P(A) / P(B).

Key Vocabulary

Sample Space	The set of all possible outcomes of a random experiment. For example, the sample space when rolling a die is {1, 2, 3, 4, 5, 6}.
Event	A specific outcome or a set of outcomes within the sample space. For example, rolling an even number on a die is an event.
Theoretical Probability	The probability of an event occurring based on mathematical reasoning and the properties of the situation, calculated as (favorable outcomes) / (total possible outcomes).
Experimental Probability	The probability of an event occurring based on the results of an actual experiment or simulation, calculated as (frequency of the event) / (number of trials).
Random Experiment	An action or process that has uncertain outcomes, where each outcome is well-defined. Examples include flipping a coin or drawing a card.