Probability & Sample Spaces
Match Probability to Value
Draw a line to match each event to its probability value.
Likelihood Scale
Sort each event on the likelihood scale.
Calculate Theoretical Probability
Circle the correct probability.
P(rolling a 3 on a standard die)
P(drawing a heart from 52 cards)
P(picking a blue from bag of 3 red, 7 blue)
P(flipping tails)
List the Sample Space
Draw a line to match each experiment to the size of its sample space.
Complementary Events
Find the probability of the complement (the event NOT happening).
P(rain) = 0.3. P(no rain) = ?
P(winning) = 1/5. P(not winning) = ?
P(rolling a 6) = 1/6. P(not rolling a 6) = ?
Predicted vs Observed
Sort each statement as about theoretical probability or experimental probability.
Design an Experiment
Plan a simple probability experiment.
You have a bag with 4 red counters and 6 blue counters. What is the theoretical probability of picking red?
If you picked a counter 50 times (replacing each time), how many times would you expect to pick red?
You actually pick red 28 times in 50 trials. Is this close to what you predicted? Does this mean the experiment is unfair?
Probability Experiment at Home
Try these probability experiments and record your results.
- 1Flip a coin 20 times. Record heads and tails. How close was your result to the theoretical 1/2?
- 2Roll a die 30 times. Record each number. Which number came up most? Does this match theoretical probability?
- 3Draw cards from a shuffled pack (10 cards: 5 red, 5 black). Record your picks. Compare to the theoretical probability.
- 4Use a spinner (draw one with 4 equal sections). Spin 40 times. Compare results to theoretical probability.
Venn Diagram — Two Events
In a class of 30, 18 play sport, 12 do art, and 5 do both. Draw a line to match each region to its value.
Mutually Exclusive Events
Two events are mutually exclusive if they cannot happen at the same time.
Are 'rolling a 3' and 'rolling a 5' mutually exclusive on one die roll?
P(A) = 0.4, P(B) = 0.3, A and B are mutually exclusive. P(A or B) = ?
A card is drawn. Are 'red card' and 'king' mutually exclusive?
If events A and B are mutually exclusive, P(A and B) = ?
Probability in Three Forms
Draw a line to match each probability fraction to its equivalent decimal and percentage.
Relative Frequency from Experiment Results
Relative frequency = number of times event occurred ÷ total number of trials.
Heads appeared 38 times in 100 coin flips. Relative frequency of heads?
Red was drawn 12 times in 40 draws. Relative frequency of red?
A 6 appeared 8 times in 60 rolls. Relative frequency of rolling 6?
Venn Diagrams and Combined Events
Use Venn diagrams to answer each question.
In a group of 25 people, 15 drink tea, 10 drink coffee, and 4 drink both. How many drink only tea? Only coffee? Neither?
A die is rolled once. Event A: rolling an even number. Event B: rolling a number > 3. List the outcomes in each region of the Venn diagram: only A, only B, both A and B, neither.
Consolidating — List the Sample Space
List all outcomes in the sample space for each experiment.
Flipping two coins. List all outcomes as ordered pairs (e.g. HH, HT, TH, TT). How many outcomes?
Rolling two dice and recording the sum. What sums are possible? List the minimum, maximum, and most common sum.
Choosing a vowel and a consonant from the word MATHS. List all pairs.
Consolidating — Sample Space Size
Circle the total number of outcomes in the sample space.
Flipping three coins
Rolling one die and flipping one coin
Choosing one from {A, B, C} and one from {1, 2}
Choosing 1 marble from a bag of 5 different colours
Consolidating — Match Event to Probability
Draw a line to match each event to its probability.
Consolidating — Expected Frequency
Use theoretical probability to calculate expected frequency.
A die is rolled 120 times. How many times would you expect to roll a 5? Show your calculation.
A bag has 2 red, 3 blue, 5 yellow. You draw (and replace) 50 times. How many times do you expect each colour?
You flip a coin 200 times. You get 112 heads. Calculate the experimental probability of heads. Is the coin fair? Explain.
Consolidating — Sort Events by Probability
Sort these events from least likely (left) to most likely (right).
Consolidating — Tree Diagrams for Two Events
Draw a tree diagram and use it to find probabilities.
A bag has 3 red and 2 blue balls. You draw one ball, note its colour, replace it, then draw again. Draw a tree diagram showing all possible outcomes.
Use your tree diagram to find P(both red), P(both blue), P(one of each colour).
Consolidating — Simulation
Use a simulation to estimate probability.
Roll a die 30 times and record results in a tally table. Calculate the experimental probability of each number 1–6.
Which number appeared most often? Which least? Do your results match theoretical probability? Why might they differ?
If you repeated this experiment 10 000 times, would the probabilities be closer to 1/6 each? Explain why.
Consolidating — Probability of Combined Events
Circle the correct probability for each combined event.
P(head AND rolling a 3): flipping a coin and rolling a die independently
P(A or B) where P(A) = 0.2, P(B) = 0.5, and A and B are mutually exclusive
A bag has 1 red, 2 blue, 3 green. P(not green) = ?
Consolidating — Event Language
Draw a line to match each probability term to its correct meaning.
Consolidating — Designing a Fair Game
Design a probability game that is fair for two players.
Player A wins if the sum of two dice is even. Player B wins if the sum is odd. Calculate the probability of each player winning. Is the game fair?
Design your own two-player game using a coin and a die. Describe the rules, calculate each player's probability of winning, and confirm the game is fair.
Consolidating — Probability in Sport
Apply probability reasoning to sporting contexts.
A basketball player makes 7 out of every 10 free throws. Write this as a probability. How many free throws would you expect in 50 attempts?
In a tennis match, Player A wins 60% of points on serve. What is the probability Player A loses their serve on any given point?
If a cricket team has won 12 of their last 20 matches, what is their experimental probability of winning the next match? Is this a good estimate? Why?
Consolidating — Interpret an Experiment
Interpret the results of a probability experiment.
A spinner has 4 equal sections labelled 1, 2, 3, 4. After 40 spins the results are: 1 → 8 times, 2 → 12 times, 3 → 9 times, 4 → 11 times. Calculate the experimental probability for each number.
Compare the experimental probabilities to the theoretical probability of 1/4. Which number's experimental probability was furthest from theoretical? Does this mean the spinner is unfair? Explain.
How many times would you expect each number to appear if you did 400 spins instead?
Extending — Tree Diagrams for Three Events
Extend your tree diagram skills to three-stage experiments.
You flip a coin three times. Draw a tree diagram showing all 8 outcomes.
Find P(exactly 2 heads in 3 flips). List all favourable outcomes.
Find P(at least 1 head). Use the complement: P(at least 1 head) = 1 − P(no heads).
Extending — Dependent Events (Without Replacement)
Find probabilities when items are NOT replaced after being drawn.
A bag has 5 red and 3 blue balls. You draw 2 balls without replacement. Find P(both red). Show the tree diagram or calculation.
Find P(first red AND second blue) without replacement.
Compare: P(both red) with replacement vs without replacement. Which is larger? Why?
Extending — The Birthday Problem
Investigate a famous probability puzzle.
In a group of 23 people, what is the probability that at least two share a birthday? Most people guess low — but it is approximately 50.7%! Explain why this might be surprising.
To find P(at least two share a birthday), it is easier to find P(no two share a birthday). P(all birthdays different for 2 people) = 365/365 × 364/365. For 3 people, add another factor. Write the expression for 4 people.
Why does the probability reach 50% with only 23 people when there are 365 possible birthdays?
Extending — Conditional Probability Introduction
Conditional probability is the probability of an event GIVEN that another event has occurred.
A bag has 4 red and 6 blue balls. You draw one ball and it is red (you know this). Now you draw a second ball without replacing the first. What is the probability the second ball is red?
Out of 100 students, 60 play sport, 40 play music, and 20 do both. Given that a student plays sport, what is the probability they also play music?
Extending — Sort by Type of Probability Event
Sort each situation into the correct category.
Extending — Monte Carlo Simulation
Monte Carlo methods use repeated random experiments to estimate probabilities.
Describe how you could use 100 coin flips to estimate the probability of getting exactly 2 heads in 3 flips. What would you record and how would you calculate the estimate?
A needle of length 1 cm is dropped randomly onto a page of parallel lines 2 cm apart. This is called Buffon's Needle — the probability of crossing a line is 1/π ≈ 0.318. If you dropped the needle 100 times and it crossed a line 31 times, what is your estimate of π from this experiment? (Hint: rearrange.)
Extending — Probability and Risk
Apply probability to everyday risk and decision-making.
A medical test for a disease is 95% accurate. The disease affects 1% of the population. Explain why a positive test result does NOT mean you definitely have the disease. (This is the 'false positive paradox'.)
An insurance company insures 10 000 cars. The probability of a car being in an accident in a year is 0.03, and the average accident costs $5 000. How much should the company expect to pay out per year? What minimum premium per car covers this?
Extending — Expected Value
Expected value is the average outcome over many trials.
A game: roll a die. If you roll 6, you win $5. If you roll 1, you win $2. Otherwise you win nothing. What is the expected value per roll?
A raffle has 100 tickets at $2 each. First prize is $50, second prize is $20. Is it worth buying a ticket? Calculate the expected return and profit/loss per ticket.
Extending — Probability Project: Analyse a Game
Analyse a game of chance using probability theory.
Choose a simple game of chance (e.g. noughts and crosses with random moves, a dice-based board game, snakes and ladders). Describe the game and identify at least 3 probability events within it.
Calculate the probability of each event you identified. Are any events complementary, mutually exclusive, or independent?
Is the game 'fair'? Does each player have an equal probability of winning? What would need to change to make it perfectly fair?
Extending — Probability and Genetics
Genetics uses probability to predict traits in offspring.
A Punnett square for two brown-eyed parents (both Bb, where B is dominant) shows offspring BB, Bb, Bb, bb. What is the probability of a child having brown eyes (BB or Bb)?
What is the probability of blue eyes (bb) in this cross?
If this couple has 3 children, what is the probability all 3 have blue eyes? Use the multiplication rule.
Extending — Statistical vs Mathematical Probability
Compare two approaches to measuring probability.
A coin is flipped 1000 times and shows heads 519 times. Write the experimental probability as a decimal. Is the coin fair? How many standard deviations is 519 from the expected 500? (Hint: SD for this experiment ≈ √(1000 × 0.5 × 0.5) ≈ 15.8.)
Can we ever prove a coin is fair using experimental data alone? Explain what 'more data' does to our confidence.
Extending — Reflection: Probability in the Real World
Summarise your understanding of probability and its applications.
List five real-world contexts where probability is used. For each, explain what is being estimated and why it matters.
Describe the difference between theoretical, experimental, and subjective probability. Give an example of each.
What is the most surprising thing you learned about probability in this worksheet? Why does it challenge your intuition?
Probability Investigations at Home
Explore probability through hands-on experiments and real-world investigation.
- 1Flip a coin 50 times. Graph cumulative proportion of heads after every 5 flips. Watch it approach 0.5.
- 2Roll two dice 36 times. Record sums. Compare experimental frequency to theoretical frequency for each sum.
- 3Collect data on something random at home (e.g. number of cars passing each minute, colour of the next 10 cars). Calculate experimental probability.
- 4Look up a weather forecast for the next 7 days. At the end of each day, record whether the prediction was correct. Calculate the forecaster's accuracy.
- 5Research how probability is used in one of: medical trials, quality control in manufacturing, financial risk modelling, or sports analytics. Write a one-page summary.
- 6Play a simple card game (e.g. snap, war) and calculate the probability of winning each round before playing. Does your theoretical prediction match real play?