Conditional Probability & Compound Events
Probability Type Match
Draw a line from each probability type to its description.
Independent vs Dependent Events
Circle whether each pair of events is independent or dependent.
Flipping a coin and rolling a die
Drawing two cards from a deck without replacement
The weather today and whether you carry an umbrella
Rolling a die twice
Choosing a team captain, then choosing a vice-captain from the remaining players
Spinning a spinner and flipping a coin
Basic Single-Event Probabilities
Circle the correct probability for each event.
P(rolling a 4 on a standard die)
P(drawing a heart from a standard 52-card deck)
P(flipping heads on a fair coin)
P(rolling an even number on a standard die)
Probability Notation Match
Draw a line from each notation to its meaning.
Independent vs Dependent Pairs
Sort each event pair into Independent or Dependent.
Sample Space Size
Circle the correct total number of outcomes in each sample space.
Rolling two standard dice
Flipping two coins
Flipping three coins
Rolling a die and flipping a coin
Drawing 1 card from a standard deck then flipping a coin
Complementary Events: P(A') = 1 − P(A)
Circle the correct complementary probability.
P(rolling a 6) = 1/6, so P(not rolling a 6) =
P(rain) = 0.3, so P(no rain) =
P(drawing a spade) = 1/4, so P(not a spade) =
P(winning a raffle) = 0.02, so P(not winning) =
'At Least One' Using the Complement
Circle the correct probability. Hint: P(at least one) = 1 − P(none).
P(at least one head in 2 coin flips). P(no heads) = (1/2)² = 1/4, so P(at least one head) =
P(at least one 6 in 2 dice rolls). P(no 6) = (5/6)² = 25/36, so P(at least one 6) =
P(at least one red marble when drawing 2 with replacement from a bag of 3 red and 7 blue). P(no red) = (7/10)² = 49/100, so P(at least one red) =
Tree Diagram — Marbles Without Replacement
A bag contains 3 red and 2 blue marbles. Two are drawn without replacement. Circle the correct probability for each outcome.
P(Red first) =
P(Blue second | Red first) =
P(Red then Blue) =
P(Both same colour) = P(RR) + P(BB) =
Reading a Two-Way Table
Use the two-way table below. 200 students were surveyed about sport and music. | | Plays Sport | No Sport | Total | |--------------|-------------|----------|-------| | Plays Music | 45 | 30 | 75 | | No Music | 80 | 45 | 125 | | Total | 125 | 75 | 200 | Circle the correct answer.
P(plays sport) =
P(plays music and plays sport) =
P(plays music | plays sport) =
P(plays sport | plays music) =
P(A and B) for Independent Events
For independent events, P(A and B) = P(A) × P(B). Circle the correct answer.
P(heads on coin 1 AND heads on coin 2) =
P(rolling a 3 AND then rolling a 5) on two dice =
A spinner has P(red) = 0.4. Two independent spins. P(red both times) =
P(rain on Monday) = 0.3, P(rain on Tuesday, independent) = 0.5. P(rain both days) =
P(A and B) for Dependent Events
For dependent events, P(A and B) = P(A) × P(B|A). Circle the correct answer.
A bag has 4 red and 6 blue marbles. P(red first, then red second without replacement) =
A deck has 52 cards. P(Ace first, then Ace second without replacement) =
A jar has 5 green and 3 yellow sweets. P(green then yellow, without replacement) =
Steps to Calculate Using the Complement
Put the steps in order to find P(at least one head in 3 coin flips).
Steps to Complete a Two-Way Table
Put the steps in order to complete a two-way table and find a conditional probability.
Scenario → Calculation Method
Draw a line from each probability scenario to the correct calculation method.
Conditional Probability from a Two-Way Table
120 Year 10 students were surveyed. | | Likes Maths | Doesn't Like Maths | Total | |--------------|-------------|--------------------|-------| | Likes Science| 35 | 15 | 50 | | No Science | 20 | 50 | 70 | | Total | 55 | 65 | 120 | Circle the correct answer.
P(likes maths | likes science) =
P(likes science | likes maths) =
P(doesn't like maths | doesn't like science) =
Are 'likes maths' and 'likes science' independent? Check: does P(maths | science) = P(maths)?
With Replacement vs Without Replacement
A bag contains 6 red and 4 blue marbles. Circle the correct probability for each scenario.
P(red then red) WITH replacement =
P(red then red) WITHOUT replacement =
P(blue then red) WITH replacement =
P(blue then red) WITHOUT replacement =
Tree Diagram Rules
Circle the correct statement about tree diagrams.
To find the probability of a specific path through a tree diagram, you:
To find the probability of multiple different outcomes (e.g., P(one red and one blue in any order)), you:
The probabilities on all branches from a single node must:
In a 'without replacement' tree diagram, the denominators on the second set of branches are:
Medical Test: False Positive Problem
Use a two-way table to solve this real-world conditional probability problem.
A medical test is 95% accurate (correctly identifies a condition when present) and has a 3% false positive rate (positive result when condition is absent). If 1 in 100 people have the condition, complete a two-way table for 10,000 people: • 100 people have the condition → 95 test positive (true positive), 5 test negative (false negative) • 9,900 don't have it → 297 test positive (false positive), 9,603 test negative (true negative) (a) Calculate P(has condition | tests positive) = 95 / (95 + 297). Show your working. (b) Why is this result surprising? Explain in plain language why a positive test doesn't mean you're likely to have the condition.
Design a Profitable Carnival Game
Apply compound probability to design and analyse a game of chance.
Design a simple carnival game using two spins of a spinner (or two draws from a bag). The game costs $2 to play. (a) Describe your game and its rules. (b) List all possible outcomes and their probabilities. (c) Assign prizes to each outcome. (d) Calculate the expected value for the player per game. (e) Show that the game operator makes a profit on average. (f) Explain why the game still feels fun and winnable despite being profitable for the operator.
The Monty Hall Problem
Analyse this famous probability puzzle using conditional reasoning.
In a game show, there are 3 doors. Behind one is a car; behind the other two are goats. You pick a door. The host (who knows where the car is) opens a different door to reveal a goat, then asks if you want to switch. (a) What is P(car behind your original door)? What is P(car behind one of the other two doors)? (b) After the host opens a goat door, what is P(car behind the remaining door)? (c) Should you switch? Explain why switching gives you a 2/3 chance of winning. (d) Many people believe it's 50/50. Explain clearly why that intuition is wrong.
Weather Probability — Dependent Events
Solve a compound probability problem involving dependent events.
In a certain city, the probability of rain on any given day is 0.3. However, if it rained the previous day, the probability of rain increases to 0.5. (a) Draw a tree diagram for two consecutive days (Day 1 and Day 2). (b) Calculate P(rain on both days) = P(rain Day 1) × P(rain Day 2 | rain Day 1) = 0.3 × 0.5. Show your working. (c) Calculate P(rain on exactly one of the two days). Consider both paths: rain-then-dry and dry-then-rain. (d) Calculate P(no rain on either day) = P(dry Day 1) × P(dry Day 2 | dry Day 1) = 0.7 × 0.7. Verify that all four outcome probabilities sum to 1.
Quality Control — Two Machines
Apply conditional probability to a manufacturing scenario.
A factory has two machines. Machine A produces 60% of all items and has a 4% defect rate. Machine B produces 40% of items and has a 6% defect rate. (a) Complete a two-way table for 1,000 items: • Machine A: 600 items → 24 defective, 576 good • Machine B: 400 items → 24 defective, 376 good (b) What is the overall defect rate? (48/1000 = 4.8%) (c) If an item is found to be defective, what is the probability it came from Machine A? Calculate P(Machine A | defective) = 24/48 = 1/2. (d) Is P(defective | Machine A) the same as P(Machine A | defective)? Explain why or why not.
True or False — Probability Properties
Circle True or False for each statement about probability.
P(A|B) is always equal to P(B|A)
If A and B are independent, then P(A|B) = P(A)
P(A ∪ B) = P(A) + P(B) is always true
P(A ∩ B) ≤ P(A) and P(A ∩ B) ≤ P(B)
The probability of any event is between 0 and 1 inclusive
For dependent events: P(A and B) = P(A) × P(B)
Sports Probability — Best-of-3 Series
Calculate compound probabilities for a sports scenario.
Team A has a 0.6 probability of winning any individual game against Team B (assume games are independent). (a) In a best-of-3 series (first to 2 wins), list all possible sequences of outcomes (e.g., AA, ABA, BAA, ABB, BAB, BB). (b) Calculate the probability of each sequence. • P(AA) = 0.6 × 0.6 = 0.36 • P(ABA) = 0.6 × 0.4 × 0.6 = 0.144 • P(BAA) = 0.4 × 0.6 × 0.6 = 0.144 • P(ABB) = 0.6 × 0.4 × 0.4 = 0.096 • P(BAB) = 0.4 × 0.6 × 0.4 = 0.096 • P(BB) = 0.4 × 0.4 = 0.16 (c) P(Team A wins series) = 0.36 + 0.144 + 0.144 = 0.648. Verify P(Team B wins) = 0.352 and check they sum to 1. (d) Is Team A's series advantage (0.648) larger or smaller than their single-game advantage (0.6)? Explain why.
Why P(A|B) ≠ P(B|A)
Explain a key concept in conditional probability with a clear example.
Using this data about 200 people: | | Left-handed | Right-handed | Total | |--------------|-------------|--------------|-------| | Plays guitar | 8 | 32 | 40 | | No guitar | 12 | 148 | 160 | | Total | 20 | 180 | 200 | (a) Calculate P(left-handed | plays guitar) = 8/40 = 1/5 = 0.2 (b) Calculate P(plays guitar | left-handed) = 8/20 = 2/5 = 0.4 (c) These are clearly different! Explain in your own words why P(A|B) ≠ P(B|A) in general. (d) Give a real-life example where confusing P(A|B) with P(B|A) could lead to a wrong conclusion.
Probability Experiments at Home
Explore conditional and compound probability with hands-on experiments.
- 1Card experiment: Draw 2 cards without replacement from a standard deck, 30 times. Record how often both are the same suit. Compare your result to the theoretical probability: P(same suit) = 13/52 × 12/51 × 4 suits = 4/17 ≈ 0.235.
- 2Monty Hall experiment: Use 3 cups and a small object. Have a family member play the host. Do 30 rounds: always switch for 15 rounds, always stay for 15 rounds. Record your wins for each strategy. Does your data support the theoretical 2/3 win rate for switching?
- 3Dice experiment: Roll two dice 50 times. Record how often you get at least one 6. Compare to the theoretical P(at least one 6) = 1 − (5/6)² = 11/36 ≈ 0.306. Discuss why your experimental result might differ from the theory.
Real-World Conditional Probability
Find and analyse conditional probability in everyday life.
- 1Research a real-world example of conditional probability (e.g., medical testing accuracy, weather forecasting, spam email filtering). Explain what the 'condition' is and why P(A|B) ≠ P(B|A) matters in that context.
- 2Create a two-way table using data from your family or class (e.g., favourite subject vs favourite sport, pet owners vs birth month). Calculate at least two conditional probabilities from your table and explain what they mean.
- 3Find a news article that mentions a probability or statistic. Identify whether it involves simple, compound, or conditional probability. Explain whether the article interprets the probability correctly.
Probability Basics — Review and Extension
Review fundamental probability concepts and extend to Year 10 contexts.
A bag contains 5 red, 3 blue, and 2 green marbles. A marble is drawn at random. Calculate: (a) P(red) (b) P(not green) (c) P(red or blue) (d) Two marbles are drawn without replacement. P(both red) (e) P(first red, second blue)
Probability Notation
Draw a line from each probability notation to its meaning.
Mutually Exclusive Events
Circle the correct answer about mutually exclusive events.
Events A and B are mutually exclusive. P(A ∪ B) =
Rolling a 3 and rolling a 5 on a single die are:
If P(A) = 0.4, P(B) = 0.3 and A, B are mutually exclusive, P(A ∪ B) =
Getting a head and getting a tail on a single coin flip are:
Tree Diagrams — Multi-Stage Events
Use a tree diagram to calculate compound probabilities.
A box has 4 red and 6 blue balls. Two balls are drawn without replacement. (a) Draw a complete probability tree diagram. (b) Calculate P(both red). (c) Calculate P(exactly one of each colour). (d) Calculate P(at least one blue). (e) Verify that all branch probabilities at the second stage sum to 1.
Conditional Probability — Bayes' Thinking
Apply conditional probability reasoning to real-world contexts.
A disease affects 1% of a population. A test for the disease is 95% accurate for those who have it, and 90% accurate for those who don't (10% false positive rate). In a group of 1,000 people: (a) How many have the disease? (b) Of those, how many test positive? (c) Of those without the disease, how many test positive? (d) If a person tests positive, what is the probability they actually have the disease? (e) Explain why this result surprises most people.
Independent vs Dependent Events
Sort each pair of events: Independent or Dependent.
Venn Diagrams — Two Events
Use Venn diagrams to organise probability calculations.
In a class of 30 students: 18 study French, 14 study Spanish, 6 study both. (a) Draw a Venn diagram. (b) How many study French only? Spanish only? Neither? (c) Find P(French | Spanish) — given a student studies Spanish, what is the probability they also study French? (d) Are studying French and studying Spanish independent events? Show mathematically.
Sample Space and Complementary Events
Circle the correct use of the complement rule.
P(at least one head in 3 flips) = 1 − P(no heads). P(no heads) =
P(at least one 6 in two dice rolls) = 1 − P(no 6). P(no 6 in two rolls) =
P(not red) when P(red) = 0.35 is:
Expected Value
Calculate and interpret expected value in probability contexts.
A game costs $5 to play. You roll a die: if you roll a 6, you win $20; if you roll a 5, you win $10; otherwise you win nothing. (a) Calculate the expected winnings from one game. (b) Calculate the expected profit/loss per game (including the $5 cost). (c) Is this a fair game? Would you play it? Justify mathematically.
An insurance company charges $300/year for a policy. They estimate a 2% chance of paying out $8,000 and a 0.5% chance of paying out $50,000 in any given year. Calculate the company's expected profit per policy per year.
Steps to Solve a Conditional Probability Problem
Put the steps in the correct order.
Probability Rules — Formula to Name
Draw a line from each probability formula to its correct name.
Simulations in Probability
Design and interpret probability simulations.
Describe how you would use a random number generator to simulate 1,000 trials of drawing 2 cards (with replacement) and counting how often both cards are the same suit. What result would you expect theoretically? [P(same suit) = 4 × (13/52)² = 4 × (1/4)² = 1/4]
Explain why simulations are useful in probability, especially for complex events where the theoretical calculation is difficult. Give an example of a probability problem (e.g. the birthday problem) where simulation is practical but exact calculation is complex.
Two-Way Tables and Conditional Probability
Extract conditional probabilities from two-way tables.
A survey of 500 people asked whether they exercise regularly and whether they have high blood pressure: | | High BP | Normal BP | Total | |------------------|---------|-----------|-------| | Exercises | 20 | 230 | 250 | | Does not exercise| 80 | 170 | 250 | | Total | 100 | 400 | 500 | (a) P(high BP | exercises) (b) P(high BP | does not exercise) (c) P(exercises | high BP) (d) Are exercise and blood pressure independent? Show mathematically.
Probability in Games and Gambling
Investigate probability in everyday games.
- 1Research the probability of winning in a common card or board game (e.g. Blackjack, Snakes and Ladders, Monopoly). Calculate at least one key probability and explain how it affects strategy.
- 2Explore the Australian lottery (e.g. Lotto). Find the probability of winning the jackpot. Calculate the expected value of a ticket if the jackpot is $5 million and a ticket costs $1.35. Is playing the lottery a good financial decision?
- 3Play 30 rounds of a simple coin/dice game with a family member. Record results. Calculate experimental probability. Compare to theoretical probability. Discuss why they may differ.
Permutations and Combinations — Introduction
Apply counting principles to probability problems.
How many different 3-letter arrangements can be made from the letters A, B, C, D, E (no repeats)? Explain the difference between a permutation (order matters) and a combination (order doesn't matter). Calculate how many 3-letter combinations (not arrangements) can be chosen.
A committee of 3 people is chosen from a group of 8. Using combinations (C(8,3) = 8!/(3!×5!)), calculate the number of possible committees. What is the probability that two specific people (Ali and Ben) are both on the committee?
Probability Values — Possible or Impossible?
Circle whether each probability value is Possible or Impossible.
P(A) = 1.2
P(A) = 0
P(A) + P(A') = 1.5
P(A|B) = 0.7 when P(A) = 0.3
Probability Rules — Match the Formula
Draw a line from each probability rule to its formula.
Two-Way Tables and Probability
Use a two-way frequency table to calculate probabilities.
A school surveyed 200 students about sport preference. 80 males: 50 like football, 30 like swimming. 120 females: 40 like football, 80 like swimming. Construct a two-way table.
Using your table, find: (a) P(female) (b) P(likes football) (c) P(male and likes swimming) (d) P(likes swimming | female)
Are sport preference and gender independent in this dataset? Justify using probabilities.
Classify Events as Independent or Dependent
Sort each pair of events as independent or dependent.
Tree Diagrams for Multi-Stage Events
Use tree diagrams to calculate compound probabilities.
A bag contains 3 red and 2 blue marbles. Draw a tree diagram for drawing two marbles without replacement. Label all branches with probabilities.
Using your tree diagram, find: (a) P(both red) (b) P(both blue) (c) P(one of each colour)
How would the tree diagram change if the first marble was replaced before drawing the second? Recalculate the probabilities.
Probability Concepts Used This Week
Tally each probability concept you applied during this unit.
| Item | Tally | Total |
|---|---|---|
Complementary probability | ||
Conditional probability | ||
Tree diagrams | ||
Two-way tables | ||
Venn diagrams |
Bayes' Theorem — Introduction
Apply Bayes' theorem to update probabilities based on new information.
State Bayes' theorem: P(A|B) = P(B|A) × P(A) / P(B). Explain each term in the formula.
A medical test for a disease is 95% accurate. The disease affects 1% of the population. If someone tests positive, find the probability they actually have the disease. Show full working using Bayes' theorem.
Why is the result of the medical test example surprising? What does it tell us about the importance of base rates?
Expected Value and Fairness
Calculate expected values and determine if games are fair.
Define expected value (E) for a probability distribution. Write the general formula.
A game costs $5 to play. You win $20 with probability 0.15, $5 with probability 0.30, and $0 otherwise. Calculate the expected profit (or loss) per game.
Is the game 'fair'? Explain what fairness means in terms of expected value.
Design your own fair game with three outcomes. Show that the expected value equals the entry cost.
Probability in Everyday Life
Explore how probability and risk affect real decisions.
- 1Research how insurance companies use probability to set premiums. Write a paragraph explaining the concept of 'expected loss'.
- 2Play a board game that involves dice or cards with a family member. Record outcomes and compare experimental probability with theoretical probability over 30 trials.
- 3Find an example of conditional probability in a news article (e.g. medical screening, weather forecasting). Explain the probabilities used.
- 4Research the 'Birthday Paradox' — the probability that two people in a group share a birthday. Calculate the probability for a group of 23 people.
- 5Investigate how poker players or chess players use probability and expected value to make decisions. Write a short summary of the strategies used.
Simulating Probability Experiments
Use simulation to estimate probabilities and compare with theory.
Describe how you could use a random number generator to simulate rolling a fair die 100 times. What would you record?
Simulate flipping a coin 50 times using a random method (coin, calculator, or app). Record your results and calculate the experimental probability of heads.
Compare your experimental result to the theoretical probability of 0.5. Calculate the absolute difference and explain whether you expected this level of variation.
How would increasing the number of trials (e.g. to 1000) affect the experimental probability? Explain using the Law of Large Numbers.
Venn Diagrams and Set Notation
Use Venn diagrams and set notation to organise and calculate probabilities.
In a group of 30 students: 18 play sport, 12 play music, 7 play both. Draw a Venn diagram with all four regions labelled.
Using the Venn diagram: find P(sport only), P(music only), P(sport or music), P(neither).
Write the addition rule in set notation: P(A ∪ B) = ... and verify it using your Venn diagram.
Mutually Exclusive and Exhaustive Events
Identify and work with mutually exclusive and exhaustive events.
Define mutually exclusive events. Give two examples from a deck of cards.
Define exhaustive events. Give an example of a set of exhaustive events when rolling a die.
Are 'drawing a heart' and 'drawing a red card' mutually exclusive? Explain and calculate P(heart or red).
If events A and B are mutually exclusive and exhaustive, what must P(A) + P(B) equal? Why?
Probability Value — Classify the Event
Sort each event by its approximate probability.
Permutations and Combinations Introduction
Begin exploring counting methods for probability calculations.
Explain the difference between a permutation and a combination. When does order matter?
How many ways can 5 students be arranged in a row for a photo? Show the calculation.
A committee of 3 is chosen from 8 people. How many possible committees are there? Use the combination formula.
A PIN uses 4 different digits from 0–9. How many different PINs are possible? Compare this to allowing repeated digits.