Compound Events & Probability
Probability Vocabulary
Draw a line from each term to its correct meaning.
List the Sample Space
Write out all possible outcomes for each compound experiment.
A coin is tossed and a 6-sided die is rolled. List all outcomes in the form (H, 1), (H, 2), ...
Two coins are tossed at the same time. List all outcomes.
A spinner with sections Red, Blue, Green is spun and a coin is tossed. List all outcomes.
Two-Way Table
A fair coin is tossed and a 4-sided die (1, 2, 3, 4) is rolled. Complete the table and answer the questions.
H row: (H,1) (H,2) (H,3) (H,4) T row: ___ ___ ___ ___ Total outcomes = ___
How many outcomes include an even number? ___ P(Heads and even number) = ___
Tree Diagrams
Draw a tree diagram for the experiment below, then answer the probability questions.
A bag contains one red (R) and one blue (B) marble. One marble is drawn and NOT replaced. Then a second marble is drawn. Draw the tree diagram and list all outcomes.
P(Red then Blue) = ___ P(Both the same colour) = ___ P(At least one Blue) = ___
Calculate the Probability
Two 6-sided dice are rolled. Circle the correct probability for each event.
P(total = 12)
P(total = 7)
P(both dice show the same number)
Complementary Probability
Use P(A complement) = 1 - P(A) to find the probability of each complementary event.
P(rolling a 6 on one die) = 1/6. What is P(NOT rolling a 6)?
When tossing two coins, P(two heads) = 1/4. What is P(NOT two heads)?
P(winning a raffle) = 3/100. What is P(not winning)?
Probability Challenge
Answer this multi-step probability problem.
A bag contains 3 red, 2 blue and 1 yellow marble. One marble is drawn and NOT replaced. Then a second marble is drawn. (a) Draw a tree diagram or table showing all outcomes. (b) Find P(both marbles are red). (c) Find P(first red, second blue).
Design a Probability Experiment
Design and carry out a compound probability experiment at home.
- 1Using two coins, toss them 20 times. Record how often you get two heads. Compare your experimental probability to the theoretical probability of 1/4.
- 2Make a simple spinner from cardboard with 3 equal sections (Red, Blue, Green). Spin it and roll a die 30 times. Which combined outcome occurred most often?
- 3Play a board game with a family member and keep track of dice totals. After 20 rolls, compare your experimental frequencies to the theoretical probabilities.
Conditional Probability P(A given B)
Conditional probability P(A|B) means the probability of A occurring GIVEN that B has already occurred. P(A|B) = P(A and B) / P(B). Use this to solve each question.
A card is drawn from a standard 52-card deck. Given it is a heart, what is the probability it is also an ace? P(ace | heart) =
In a class of 30 students, 18 play sport and 12 play music. 6 students play both. Given a student plays sport, what is the probability they also play music?
Venn Diagrams for Two Overlapping Events
Draw a Venn diagram and use it to find probabilities.
In a group of 40 students: 25 like football, 20 like cricket, and 10 like both. Draw a Venn diagram showing the groups. How many like neither sport?
From the Venn diagram above: P(football only) = ___. P(cricket only) = ___. P(both) = ___.
Addition Rule: P(A or B)
Use P(A or B) = P(A) + P(B) - P(A and B) to solve each problem.
P(A) = 0.5, P(B) = 0.4, P(A and B) = 0.2. Find P(A or B).
In a group of 50 people, 30 drink coffee, 20 drink tea, and 10 drink both. Find P(a randomly chosen person drinks coffee or tea).
If P(A) = 3/8, P(B) = 1/4 and the events are mutually exclusive (cannot both occur), find P(A or B).
Expected Value Concept
The expected value (expected outcome) for a probability experiment is calculated by multiplying each outcome by its probability and adding the results.
A game pays $10 if you roll a 6 on a die, and $0 otherwise. What is the expected payout per roll? (Hint: Expected value = 10 x P(6) + 0 x P(not 6))
A spinner has sections: win $5 (P = 0.2), win $2 (P = 0.5), lose $1 (P = 0.3). What is the expected value per spin? Is this game worth playing?
Complementary Events
Use the complement rule to find probabilities.
The probability of rain today is 0.35. What is the probability it does NOT rain?
P(rolling a 1 or 2 on a die) = 1/3. What is P(not rolling a 1 or 2)?
A factory has a 2% defect rate. If you buy one item, what is the probability it is NOT defective?
Explain why P(A) + P(not A) = 1 always holds. What does this mean geometrically?
Two-Dice Probability Table
Use a table to find probabilities for rolling two dice.
Draw and complete a two-dice sum table. How many cells in total?
Use your table to find P(sum = 7).
Find P(sum = 12).
Find P(sum > 9).
Which sum is most likely? Why?
Tree Diagrams — Two-Stage Experiments
Draw and use tree diagrams to find probabilities.
A bag has 4 red (R) and 2 green (G) balls. Draw a tree diagram for drawing two balls WITH replacement.
Use your tree diagram to find P(R, R), P(R, G), P(G, R), P(G, G). Do the probabilities add to 1?
What is P(at least one green ball)?
Without Replacement — Dependent Events
Calculate probabilities for two-stage experiments without replacement.
A bag has 5 red and 3 blue balls. You draw two balls WITHOUT replacement. Draw a tree diagram.
Find P(red, then red).
Find P(one of each colour) — i.e. P(red, then blue) OR P(blue, then red).
Compare your answers to the WITH replacement case. Which is more likely to give two reds — with or without replacement? Why?
Probability Rules — Match Formula to Name
Match each rule to its formula.
Venn Diagrams and Probability
Use Venn diagrams to organise and calculate probabilities.
In a class of 30, 18 play sport, 12 do art, and 6 do both. Draw a Venn diagram. How many do sport only? Art only? Neither?
Find P(sport), P(art), P(sport and art), P(sport or art), P(neither).
Verify that P(sport or art) = P(sport) + P(art) − P(sport and art).
Probability in Everyday Life
Apply probability thinking to real-world situations.
A test for a disease is 95% accurate. If the disease affects 1 in 1000 people, explain why a positive test result might NOT mean you are sick. (This is the false positive paradox.)
A car insurance company charges more for young drivers. Use probability thinking to explain why this is mathematically justified, even if it seems unfair to safe young drivers.
If you flip a fair coin 10 times and get 10 heads, what is the probability the next flip is heads?
Conditional Probability — Introduction
Explore conditional probability P(A|B) — the probability of A given B has occurred.
A class has 15 boys and 10 girls. 6 boys and 4 girls play chess. A student is chosen at random. Given the student is a girl, find P(plays chess).
From a standard deck, a card is drawn. Given it is red, what is P(it is a heart)?
Explain the difference between P(A and B) and P(A|B). When would these be equal?
Simulation and Experimental Probability
Use simulation to estimate probability.
Flip a coin 30 times and record results. Calculate your experimental P(heads). How close is it to 0.5?
Roll a die 60 times. Tally the results. Calculate experimental probability for each face.
If your experimental probability differs from the theoretical value, does that mean the coin/die is unfair? Explain.
Problem Solving — Multi-Step Probability
Solve these multi-step probability problems.
A box contains 6 white, 4 black, and 2 red marbles. Two marbles are drawn with replacement. Find: (a) P(both white), (b) P(one white, one black), (c) P(at least one red).
Three students independently try to solve a problem. Their probabilities of success are 0.7, 0.6, and 0.5 respectively. What is the probability that at least one of them solves it?
Probability in Daily Life
Investigate probability in real contexts.
- 1Track the weather forecast for your area for 2 weeks. Record the predicted probability of rain and whether it actually rained. Compare the forecasts to reality — how accurate were they?
- 2Look at the labels on 5 different food products. Find the serving size and calculate probabilities such as 'P(a randomly chosen 10 g serving has more than 2 g of fat)'.
- 3Research the rules of the Australian Lottery. Calculate the probability of winning first prize. Compare this to the probability of other rare events (being struck by lightning, etc.).
Compound Probability — Design a Game
Design a probability game.
Design a two-stage probability game using coins, dice, or cards. Describe the rules.
Draw a tree diagram for your game and calculate P(player wins).
Calculate the expected value of your game (if it involves money). Is it fair, advantageous to the player, or advantageous to the house?
How could you adjust the rules or prizes to make it exactly fair?
Probability — Reflection and Review
Reflect on your probability learning.
Explain the difference between experimental and theoretical probability. Give an example of each.
Explain in your own words why the multiplication rule applies to independent events but NOT to dependent events.
Describe one real-world situation where understanding compound probability would help you make a better decision.
What did you find most interesting or challenging about probability? What are your next steps?
Factorial and Counting
Solve the following problems.
Calculate 5! and explain what it counts.
A password uses 3 different letters from {A, B, C, D}. How many passwords are possible?
How many ways can 4 students line up for a photo?
How does counting help us calculate theoretical probability?
Probability Terms — Match
Match each term to its meaning.
Which Formula?
Circle the correct formula to use.
Two independent events — probability both occur:
Two mutually exclusive events — probability at least one occurs:
Probability event does NOT occur:
Probability of A given B has already occurred:
Card Probability Problems
A standard deck has 52 cards: 4 suits, 13 values. Solve each problem.
What is P(drawing an ace)?
What is P(drawing a red card or a king)? Show your use of the addition rule.
Two cards are drawn without replacement. What is P(both are hearts)?
Are 'drawing a heart' and 'drawing a king' independent events? Justify.
Tree Diagram — Without Replacement
Draw and use a tree diagram.
A bag contains 4 green and 3 yellow balls. Draw a tree diagram for two draws WITHOUT replacement.
Find P(both green).
Find P(one of each colour).
How would your answers change if you replaced the ball after the first draw?
Expected Value Problems
Calculate the expected value for each scenario.
A spinner has sections worth $10 (P = 0.2), $5 (P = 0.5), and $0 (P = 0.3). Calculate the expected payout.
A raffle ticket costs $2. There is one $100 prize among 200 tickets. What is the expected value of buying one ticket?
Is the raffle in the previous question a good deal? Explain.
Design your own expected value problem and solve it.
Probability Calculations — Circle Correct
Circle the correct answer.
A bag has 5 red, 3 blue, 2 green. P(not red) = ?
P(A) = 0.6, P(B) = 0.5, A and B independent. P(A and B) = ?
P(A) = 0.3, P(B|A) = 0.4. P(A and B) = ?
P(A or B) = 0.7, P(A) = 0.4, P(B) = 0.5. P(A and B) = ?
Sort by Probability Size
Sort these events from least likely to most likely.
Two-Way Frequency Tables
Use the table to answer probability questions.
A survey of 100 students found: 40 play sport (30 pass, 10 fail), 60 don't play sport (40 pass, 20 fail). Find P(pass | plays sport).
Find P(plays sport | passes).
Are 'passing' and 'playing sport' independent? Use probabilities to justify your answer.
Construct a two-way table for a different scenario and write two conditional probability questions for it.
Probability — Mixed Problem Set
Solve each problem using appropriate methods.
A jar has 4 red, 5 blue, and 3 white balls. Two balls are drawn without replacement. Find P(both red).
P(it rains) = 0.4. P(I am late | it rains) = 0.6. P(I am late | no rain) = 0.1. Find P(I am late).
A student rolls two dice. Find P(sum > 9).
P(A) = 0.5, P(B) = 0.4, P(A and B) = 0.2. Are A and B independent? Show your working.
Probability Investigation at Home
Explore probability in everyday life.
- 1Flip a coin 50 times and record results. Calculate experimental probability of heads and compare to theoretical probability.
- 2Roll a single die 60 times. Make a tally chart. How close is each outcome to the expected 10 times?
- 3Play a card game (Snap, Go Fish) and think about which cards have been played. How does this change probabilities?
- 4Find a weather forecast. Note the probability of rain. Check back after the forecast period and track how accurate the probabilities were over a week.
- 5Research a board game involving dice or cards. Discuss the role of probability in the game strategy.
Probability Project — Experimental Study
Design and conduct a probability experiment.
Describe your experiment clearly (what you will do, what outcomes you will record).
State the theoretical probability for your key outcome and explain why.
Record your results in a table (run at least 50 trials).
Calculate your experimental probability and compare it to the theoretical value. Explain any difference.
What would happen if you ran 1000 trials? How do you know?
Medical Testing — Conditional Probability
Apply conditional probability to a medical testing scenario.
A disease affects 1% of the population. A test is 90% accurate (P(positive|disease) = 0.9, P(negative|no disease) = 0.95). If a person tests positive, what is P(they actually have the disease)?
Are you surprised by the answer? Why do you think P(disease | positive) is lower than expected?
How does the rarity of the disease affect the reliability of the test result?
Sample Space — Three Stages
Construct full sample spaces for these experiments.
List all outcomes for rolling a die and flipping a coin. How many outcomes are there?
Use a tree diagram to list all outcomes for flipping three coins.
P(exactly 2 heads from three coin flips) = ? Use your tree diagram.
Why does the total number of outcomes in a sample space depend on the number of stages?
Probability — Written Justification
Explain and justify your reasoning fully.
Explain why P(A and B) ≤ P(A) and P(A and B) ≤ P(B) for any two events.
A student says 'I rolled a 6 five times in a row, so I am unlikely to roll a 6 next time.' Explain the error in this reasoning.
Why is P(A or B) always greater than or equal to both P(A) and P(B)?
Explain how a Venn diagram represents P(A and B) and P(A or B) visually.
Probability Concepts — Sort by Complexity
Sort these probability ideas from most basic to most advanced.
Probability — Connecting to Statistics
Explore the relationship between probability and statistics.
How is theoretical probability different from statistical frequency?
A biased coin is flipped 1000 times, landing heads 620 times. Estimate P(heads) and justify your estimate.
Explain how you could use repeated experiments to test whether a coin is biased.
How might probability theory be used to test a medical treatment?
Probability — Identify the Correct Approach
Circle the most appropriate method.
Finding P(sum = 7) when rolling two dice:
Testing whether a coin is biased:
Finding P(ace | card is red):
Two events cannot both occur. P(A or B) = ?
Probability — Self-Assessment and Next Steps
Reflect on your learning in this worksheet.
List three probability concepts you feel confident about.
List two concepts you still find challenging. What steps will you take to improve?
Write one question you would like to explore further about probability.
How has your understanding of probability changed through this worksheet?
Probability Rules — Match Formula to Name
Match each rule name to its formula.
Probability — Your Own Problems
Create and solve your own probability problems.
Write a problem involving two independent events and solve it.
Write a problem involving conditional probability and solve it.
Write a problem requiring the addition rule and solve it.
Probability — Which Statement Is Correct?
Circle the correct statement.
About the complement rule:
About independent events: