Complementary events
Use the complementary event rule P(A') = 1 - P(A) to find probabilities, including problems where using the complement is the most efficient method.
Worked examples
Using the complementary event rule
Straightforward
Problem
The probability that it will rain tomorrow is . Find the probability that it will NOT rain tomorrow.
1
Identify the complementary event
The event is 'rain tomorrow'. Its complement is 'no rain tomorrow'. These two events are mutually exclusive and together cover all possible outcomes, so their probabilities must add to .
2
Apply the complementary event rule
Answer
Finding a complementary probability from a sample space
Moderate
Problem
A number is chosen at random from the integers to inclusive. Find the probability that the number chosen is NOT a multiple of .
1
Find
Multiples of from to : — that is numbers.
2
Apply the complementary event rule
Answer
At least one outcome — using the complement
Challenging
Problem
Two fair six-sided dice are rolled. Find the probability that at least one die shows a .
1
Identify the complementary event
The complement of 'at least one ' is 'no on either die'. It is much easier to calculate the complement first.
2
Find
For each die, . Since the dice are independent:
3
Apply the complementary event rule
Answer
Practise
Q1·Straightforward
The probability of an event occurring is . What is , the probability that does not occur?
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Explanation
Q2·Straightforward
The probability of event is . What is the probability that does not occur?
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Explanation
Q3·Straightforward
A fair six-sided die is rolled. What is the probability of NOT rolling a ?
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Explanation
There is outcome that gives a out of possible outcomes, so .
Q4·Straightforward
The probability that a player wins a game is . What is the probability that the player does not win?
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Explanation
Q5·Moderate
A bag contains red, blue, and yellow counters. One counter is chosen at random. Use the complementary event rule to find .
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Explanation
Total counters: . .
Q6·Moderate
The probability of event is . What is ?
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Explanation
Q7·Moderate
A letter is chosen at random from the letters of the English alphabet. The vowels are A, E, I, O and U. Using the complementary event rule, find .
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Explanation
.
Q8·Moderate
A number is chosen at random from the integers to inclusive. Using the complementary event rule, find the probability that the number is NOT a multiple of .
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Explanation
Multiples of from to : — that is numbers. .
Q9·Challenging
Three fair coins are tossed. Find the probability of getting at least one tail. (Hint: use the complementary event.)
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Explanation
The complementary event is 'all three coins show heads'. There is only such outcome (HHH) out of equally likely outcomes. .
Q10·Challenging
A number is chosen at random from the integers to inclusive. Find the probability that the number is divisible by neither nor . (Use the complementary event and inclusion–exclusion.)
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Explanation
Multiples of up to : numbers. Multiples of up to : numbers. Multiples of (both): numbers. By inclusion–exclusion: numbers are divisible by or . .
Q11·Challenging
Two fair six-sided dice are rolled. Find the probability that at least one die shows a . (Use the complementary event.)
/
Explanation
. Since the dice are independent, .
Q12·Challenging
A spinner has equal sections numbered to . Find the probability that the number selected is NOT a perfect square.
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Explanation
Perfect squares from to : — that is numbers. .
Open Math
Complementary events
Statistics and Probability · MA5-PRB-C-01
Name:
Date:
Q1Straightforward
The probability of an event occurring is . What is , the probability that does not occur?
Q2Straightforward
The probability of event is . What is the probability that does not occur?
Q3Straightforward
A fair six-sided die is rolled. What is the probability of NOT rolling a ?
Q4Straightforward
The probability that a player wins a game is . What is the probability that the player does not win?
Q5Moderate
A bag contains red, blue, and yellow counters. One counter is chosen at random. Use the complementary event rule to find .
Q6Moderate
The probability of event is . What is ?
Q7Moderate
A letter is chosen at random from the letters of the English alphabet. The vowels are A, E, I, O and U. Using the complementary event rule, find .
Q8Moderate
A number is chosen at random from the integers to inclusive. Using the complementary event rule, find the probability that the number is NOT a multiple of .
Q9Challenging
Three fair coins are tossed. Find the probability of getting at least one tail. (Hint: use the complementary event.)
Q10Challenging
A number is chosen at random from the integers to inclusive. Find the probability that the number is divisible by neither nor . (Use the complementary event and inclusion–exclusion.)
Q11Challenging
Two fair six-sided dice are rolled. Find the probability that at least one die shows a . (Use the complementary event.)
Q12Challenging
A spinner has equal sections numbered to . Find the probability that the number selected is NOT a perfect square.
Worked solutions and answers at openmath.au/year-9/probability/complementary-events