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 ?
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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?
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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 ?
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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?
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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 ?
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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?
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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?
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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 ?
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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?
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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?
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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 ?
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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?
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Explanation
Kings: . Black cards: . Black kings (both): . By inclusion–exclusion: favourable outcomes. .