Sample space
List and count all possible outcomes in a sample space for single and multi-stage experiments, and apply the multiplication principle.
Worked examples
Listing a simple sample space
Straightforward
Problem
A spinner has equal sectors numbered to . List the sample space when the spinner is spun once and state how many outcomes it contains.
1
Identify all the possible outcomes. Each sector is a separate outcome.
The spinner can land on , , , , or .
2
Write the sample space using set notation.
3
Count the number of outcomes.
There are elements in the set, so the sample space contains outcomes.
Answer
; outcomes
Two-stage experiment
Moderate
Problem
A coin is flipped and then a fair six-sided die is rolled. List all the outcomes in the sample space using a table, and state how many outcomes there are.
1
Identify the outcomes for each stage.
Coin: — outcomes. Die: — outcomes.
2
Systematically list all combinations by pairing each coin outcome with each die outcome.
3
Apply the multiplication principle to count the total outcomes.
Answer
The sample space has outcomes: .
Three-stage experiment using the multiplication principle
Challenging
Problem
A student chooses a shirt from colours (red, blue, white), a pair of pants from styles (jeans, chinos), and a pair of shoes from options. How many different outfits are possible?
1
Identify the number of choices at each stage.
Shirts: choices. Pants: choices. Shoes: choices.
2
State the multiplication principle: for a multi-stage experiment, the total number of outcomes equals the product of the number of outcomes at each stage.
3
Calculate the product.
Answer
different outfits are possible.
Practise
Q1·Straightforward
A fair six-sided die is rolled once. How many outcomes are in the sample space?
Explanation
The possible outcomes are . There are outcomes in the sample space.
Q2·Straightforward
A coin is flipped once. Which set correctly lists all possible outcomes in the sample space?
Explanation
Flipping a coin can result in Heads () or Tails (). The sample space is , which contains exactly outcomes.
Q3·Straightforward
A letter is chosen at random from the word MATH. How many outcomes are in the sample space?
Explanation
The letters in MATH are . Each letter is different, so there are outcomes in the sample space.
Q4·Straightforward
A bag contains a red, a blue, a green, and a yellow marble. One marble is drawn at random. How many outcomes are in the sample space?
Explanation
The possible outcomes are . There are outcomes in the sample space.
Q5·Moderate
A coin is flipped and then a fair four-sided die numbered to is spun. How many outcomes are in the sample space?
Explanation
The coin has outcomes and the die has outcomes . By the multiplication principle, the total number of outcomes is . The outcomes are: .
Q6·Moderate
Two coins are flipped at the same time. How many outcomes are in the sample space?
Explanation
Each coin has outcomes. The sample space is , which has outcomes.
Q7·Moderate
A student can choose a main meal from options (pasta, pizza, or salad) and a drink from options (water or juice). How many different meal combinations are in the sample space?
Explanation
There are meal choices and drink choices. By the multiplication principle, the total number of combinations is .
Q8·Moderate
A fair six-sided die is rolled and a coin is flipped. How many outcomes are in the sample space?
Explanation
The die has outcomes and the coin has outcomes. By the multiplication principle, the total is outcomes.
Q9·Challenging
A fair coin is flipped three times. How many outcomes are in the sample space?
Explanation
Each flip has outcomes. For three flips, the total number of outcomes is . The outcomes are: .
Q10·Challenging
A restaurant offers entrees, main courses, and desserts. How many different three-course meals are possible?
Explanation
By the multiplication principle, the number of different three-course meals is .
Q11·Challenging
A car is available in colours, body styles, and with or without a sunroof. How many different car configurations are in the sample space?
Explanation
The three stages have , , and options respectively. By the multiplication principle, the total is configurations.
Q12·Challenging
A password is formed by choosing one letter from , then one digit from , then one symbol from . How many different passwords are possible?
Explanation
There are letter choices, digit choices, and symbol choices. By the multiplication principle, the total number of passwords is .