Theoretical probability
Calculate the theoretical probability of simple events, complementary events, and combined events using two-way tables and tree diagrams.
Worked examples
Probability of a simple event
Straightforward
Problem
A bag contains 3 red marbles and 7 blue marbles. One marble is drawn at random. Find the probability of drawing a red marble.
1
Count the total number of outcomes.
Total marbles
2
Count the number of favourable outcomes.
There are 3 red marbles, so there are 3 favourable outcomes.
3
Apply the probability formula.
Answer
3/10
Probability of a complementary event
Moderate
Problem
A standard six-sided die is rolled. Find the probability of NOT rolling a prime number.
1
List all possible outcomes.
The outcomes are — 6 equally likely outcomes.
2
Identify the prime numbers on the die.
Prime numbers: . There are 3 primes, so .
3
Use the complementary rule to find P(not prime).
Answer
1/2
Probability from a two-way table
Challenging
Problem
A survey of 50 people records their gender and beverage preference. The results: 15 males prefer tea, 10 males prefer coffee, 12 females prefer tea and 13 females prefer coffee. A person is chosen at random. Find the probability the person is male and prefers coffee.
1
Organise the data into a two-way table.
Male: tea 15, coffee 10, total 25. Female: tea 12, coffee 13, total 25. Overall total: 50.
2
Identify the number of favourable outcomes.
The cell for male and coffee shows 10 people.
3
Apply the probability formula.
Answer
1/5
Practise
Q1·Straightforward
A standard six-sided die is rolled. What is the probability of rolling a 5?
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Explanation
There are 6 equally likely outcomes: . Only one outcome is a 5, so:
Q2·Straightforward
A fair coin is flipped. What is the probability of getting tails?
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Explanation
There are 2 equally likely outcomes: heads and tails. One outcome is tails, so:
Q3·Straightforward
Cards numbered 1 to 10 are placed in a bag. One card is drawn at random. What is the probability of drawing a card showing an even number?
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Explanation
The even numbers from 1 to 10 are — that is 5 favourable outcomes out of 10 total:
Q4·Straightforward
The letters A, B, C, D and E are each written on a separate card and placed in a bag. One card is drawn at random. What is the probability of drawing a vowel?
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Explanation
The vowels among are A and E — that is 2 favourable outcomes out of 5 total:
Q5·Moderate
A standard six-sided die is rolled. What is the probability of NOT rolling a 2?
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Explanation
The probability of rolling a 2 is . Using the complementary rule:
Alternatively, the outcomes that are not 2 are — that is 5 out of 6.
Alternatively, the outcomes that are not 2 are — that is 5 out of 6.
Q6·Moderate
A standard six-sided die is rolled. What is the probability of rolling a number less than 3 or greater than 4?
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Explanation
Numbers less than 3: . Numbers greater than 4: . These groups do not overlap, so the favourable outcomes are — that is 4 out of 6:
Q7·Moderate
A bag contains 4 red marbles, 3 blue marbles and 3 green marbles. One marble is drawn at random. What is the probability of NOT drawing a blue marble?
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Explanation
Total marbles: . Marbles that are not blue: . So:
Q8·Moderate
The letters in the word MATHEMATICS are written on separate cards and placed in a bag. One card is chosen at random. What is the probability of choosing an M or an A?
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Explanation
MATHEMATICS has 11 letters: M, A, T, H, E, M, A, T, I, C, S. The letter M appears twice and the letter A appears twice, giving 4 favourable outcomes out of 11:
Q9·Challenging
A survey of 40 students records their gender and favourite sport. The results: 12 males prefer football, 8 males prefer basketball, 6 females prefer football and 14 females prefer basketball. A student is chosen at random. What is the probability the student is female and prefers basketball?
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Explanation
From the two-way table, 14 students are female and prefer basketball. The total is 40 students:
Q10·Challenging
A survey of 30 students records their year group and whether they like maths. The results: 11 Year 7 students like maths, 4 Year 7 students do not like maths, 9 Year 8 students like maths and 6 Year 8 students do not like maths. A student is chosen at random. What is the probability the student is in Year 8 and does not like maths?
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Explanation
From the two-way table, 6 students are in Year 8 and do not like maths. The total is 30 students:
Q11·Challenging
A coin is flipped and a spinner with three equal sections labelled 1, 2 and 3 is spun. List all possible outcomes in a tree diagram. What is the probability of getting heads and landing on an odd number?
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Explanation
The 6 equally likely outcomes are: (H, 1), (H, 2), (H, 3), (T, 1), (T, 2), (T, 3). The favourable outcomes — heads and an odd number — are (H, 1) and (H, 3), giving 2 out of 6:
Q12·Challenging
A bag contains 2 red balls and 3 blue balls. A ball is drawn at random, its colour is noted, then it is replaced. A second ball is drawn. Using a tree diagram, find the probability that both balls drawn are the same colour.
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Explanation
Using the tree diagram with replacement: