Theoretical probability
Calculate the theoretical probability of events in equally-likely outcome spaces, including combined and complementary events.
Worked examples
Finding probability from a simple sample space
Straightforward
Problem
A spinner has equal sections numbered to . It is spun once. Find the probability of landing on a number less than .
1
Identify the sample space
The spinner can land on any of the numbers . There are equally likely outcomes.
2
Identify the favourable outcomes
Numbers less than : . There are favourable outcomes.
3
Write the probability as a fraction
Answer
Probability of a combined event
Moderate
Problem
A fair six-sided die is rolled. Find the probability of rolling an odd number or a number greater than .
1
List the outcomes for each condition
Odd numbers: . Numbers greater than : .
2
Find the union (all outcomes satisfying either condition)
Union: . Note that appears in both sets but is only counted once. There are favourable outcomes.
3
Calculate the probability
Answer
Probability using inclusion–exclusion
Challenging
Problem
A number is chosen at random from the integers to inclusive. Find the probability that it is a multiple of or a multiple of .
1
Find : multiples of from to
Multiples of : — that is numbers. .
2
Find : multiples of from to
Multiples of : — that is numbers. .
3
Find : multiples of both and (i.e. multiples of )
Multiples of : — that is numbers. .
4
Apply the inclusion–exclusion rule
Answer
Practise
Q1·Straightforward
A fair six-sided die is rolled once. What is the probability of rolling a ?
/
Explanation
There is outcome that gives a 4, out of equally likely outcomes. So .
Q2·Straightforward
A bag contains red marbles and blue marbles. One marble is chosen at random. What is the probability of choosing a red marble?
/
Explanation
There are red marbles out of total. So .
Q3·Straightforward
A letter is chosen at random from the word MATHS. What is the probability of choosing the letter ?
/
Explanation
The word MATHS has letters, each appearing once. So .
Q4·Straightforward
A card is drawn at random from a standard deck of cards. What is the probability of drawing a heart?
/
Explanation
There are hearts out of cards. .
Q5·Moderate
A fair six-sided die is rolled. What is the probability of rolling an even number or a number greater than ?
/
Explanation
Even numbers: . Numbers greater than : . Union: — that's outcomes. .
Q6·Moderate
A bag contains green, yellow, and purple counters. One counter is chosen at random. What is the probability that it is green or yellow?
/
Explanation
Total counters: . Favourable (green or yellow): . .
Q7·Moderate
The letters A, B, C, D, E, F, G are each written on a separate card. One card is chosen at random. What is the probability of choosing a vowel?
/
Explanation
The vowels in the list are A and E — that is out of cards. .
Q8·Moderate
A number is chosen at random from the integers to inclusive. What is the probability that it is a multiple of ?
/
Explanation
Multiples of up to : — that is values. .
Q9·Challenging
Two fair coins are tossed together. What is the probability of getting exactly one head?
/
Explanation
Sample space: — equally likely outcomes. Outcomes with exactly one head: — outcomes. .
Q10·Challenging
A fair die is rolled and a fair coin is tossed. What is the probability of rolling a number less than and getting a tail?
/
Explanation
Total outcomes: . Favourable: die shows or (2 ways) AND coin shows T (1 way) outcomes. .
Q11·Challenging
A number is chosen at random from the integers to inclusive. What is the probability that the number is a multiple of or a multiple of ?
/
Explanation
Multiples of up to : numbers. Multiples of up to : numbers. Multiples of up to (counted in both): numbers. By inclusion–exclusion: favourable outcomes. .
Q12·Challenging
A card is drawn at random from a standard deck of cards. What is the probability that it is a king or a black card?
/
Explanation
Kings: . Black cards: . Black kings (both): . By inclusion–exclusion: favourable outcomes. .
Open Math
Theoretical probability
Statistics and Probability · MA5-PRB-C-01
Name:
Date:
Q1Straightforward
A fair six-sided die is rolled once. What is the probability of rolling a ?
Q2Straightforward
A bag contains red marbles and blue marbles. One marble is chosen at random. What is the probability of choosing a red marble?
Q3Straightforward
A letter is chosen at random from the word MATHS. What is the probability of choosing the letter ?
Q4Straightforward
A card is drawn at random from a standard deck of cards. What is the probability of drawing a heart?
Q5Moderate
A fair six-sided die is rolled. What is the probability of rolling an even number or a number greater than ?
Q6Moderate
A bag contains green, yellow, and purple counters. One counter is chosen at random. What is the probability that it is green or yellow?
Q7Moderate
The letters A, B, C, D, E, F, G are each written on a separate card. One card is chosen at random. What is the probability of choosing a vowel?
Q8Moderate
A number is chosen at random from the integers to inclusive. What is the probability that it is a multiple of ?
Q9Challenging
Two fair coins are tossed together. What is the probability of getting exactly one head?
Q10Challenging
A fair die is rolled and a fair coin is tossed. What is the probability of rolling a number less than and getting a tail?
Q11Challenging
A number is chosen at random from the integers to inclusive. What is the probability that the number is a multiple of or a multiple of ?
Q12Challenging
A card is drawn at random from a standard deck of cards. What is the probability that it is a king or a black card?
Worked solutions and answers at openmath.au/year-9/probability/theoretical-probability