Practice Maths

Complementary Events

Key Ideas

Key Terms

Complement
The event that A does NOT happen, written A′. P(A′) = 1 − P(A).
Complementary events
Two events that are both mutually exclusive and exhaustive — one must occur and they cannot both occur.
Mutually exclusive
Two events that cannot happen at the same time; P(A and B) = 0.
Exhaustive
A set of events that covers all possible outcomes; their probabilities sum to 1.
P(A′)
The probability of the complement of A — read as “P of A prime” or “P of not A”.

Using the Complement Rule

It is often easier to find P(A′) and subtract from 1 than to calculate P(A) directly.
For example: P(at least one head) = 1 − P(no heads).

Hot Tip Whenever you see “at least one” or “at most…” in a probability question, think about using the complement. It is usually much quicker than listing every favourable outcome.

Worked Example

Question: A bag contains 3 red, 7 blue, and 5 green marbles. A marble is drawn at random. Find P(not blue).

Step 1 — Find P(blue).
Total = 15 marbles; P(blue) = 7/15

Step 2 — Apply the complement rule.
P(not blue) = 1 − 7/15 = 8/15

Check: P(blue) + P(not blue) = 7/15 + 8/15 = 15/15 = 1 ✓

What Are Complementary Events?

Every event A has a complement, written as A' (read "A prime" or "not A"). The complement is the event that A does not happen — every outcome that is not in A. Together, A and A' cover every possible outcome, so their probabilities must add to exactly 1.

The rule is simple: P(A') = 1 − P(A)

Example: If the probability of rain tomorrow is 0.3, then the probability of no rain is 1 − 0.3 = 0.7. If the probability of rolling an odd number on a die is 3/6 = 1/2, then the probability of rolling an even number (the complement) is also 1/2.

Why Complements Are Useful

Sometimes it is much easier to calculate the probability of an event not happening and then subtract from 1, rather than calculating the event directly. This is especially true for problems that say "at least one" — because calculating "at least one success" directly can be complicated, but calculating "zero successes" (the complement) is usually straightforward.

Example: What is the probability of getting at least one Head when flipping a coin twice? You could list all outcomes: {HH, HT, TH, TT} — 3 out of 4 contain at least one Head, so P(at least one Head) = 3/4. But alternatively: P(no Heads) = P(TT) = 1/4, so P(at least one Head) = 1 − 1/4 = 3/4. Same answer, but the complement method becomes much more powerful for harder problems.

Recognising When to Use the Complement

Watch for these phrases in probability questions — they are strong signals that using the complement will make your life easier:

  • "At least one ..." (complement is "none at all")
  • "At least two ..." (complement is "zero or one")
  • "Not all ..." (complement is "all of them")
  • "One or more ..." (complement is "none")

Whenever the event you want has many possible ways to happen but the complement has only one or two ways to happen, use the complement. It saves a lot of work.

Common Mistakes with Complements

The biggest mistake is confusing the complement with something else. For example, the complement of "rolling a 6" is NOT "rolling a 1" — it is "rolling anything that is not a 6" (i.e. rolling 1, 2, 3, 4, or 5). The complement covers ALL other outcomes, not just the "opposite" one.

Another common error: forgetting that P(A) + P(A') = 1 exactly. Sometimes students add the two probabilities and get something slightly different (e.g. 0.99) because they rounded along the way. Try to keep fractions exact until the final step.

Worked Example

In a bag there are 3 red, 5 blue, and 2 green marbles. A marble is drawn at random. What is the probability of not drawing a blue marble?

P(blue) = 5/10 = 1/2.
P(not blue) = 1 − 1/2 = 1/2.

Check: P(not blue) = P(red or green) = (3 + 2)/10 = 5/10 = 1/2. Confirmed!

Key tip: The complement rule is one of the most powerful shortcuts in probability. Whenever a question asks for "at least one" or "not all," try calculating the complement first. P(at least one) = 1 − P(none). Always verify your answer makes sense: P(A) and P(A') should add to exactly 1.

Mastery Practice

  1. Use the complement rule P(A′) = 1 − P(A) to find the missing probability. Fluency

    1. P(A) = 0.3, find P(A′).
    2. P(B) = 3/8, find P(B′).
    3. P(C) = 72%, find P(C′).
    4. P(A′) = 0.45, find P(A).
    5. P(B′) = 5/9, find P(B).
    6. P(A′) = 15%, find P(A).
    7. P(A) = 1, find P(A′).
    8. P(A) = 0, find P(A′).

  2. Find the probability of the complementary event. Express answers as fractions in simplest form. Fluency

    A standard six-sided die is rolled once.

    1. P(not a 5)
    2. P(not an even number)
    3. P(not a factor of 6)
    4. P(not greater than 4)

    A card is drawn from a standard deck of 52 cards.

    1. P(not a heart)
    2. P(not a king)
    3. P(not a face card)
    4. P(not a red card)

  3. A bag contains 5 red, 3 blue, 4 green, and 8 yellow marbles. Calculate each probability using the complement where appropriate. Fluency

    1. P(not red)
    2. P(not blue)
    3. P(not yellow)
    4. P(not green)
    5. P(not red and not blue)
    6. P(red or blue)

  4. For each pair of events, state whether they are: (A) mutually exclusive only, (B) complementary (mutually exclusive AND exhaustive), or (C) neither. Understanding

    1. Rolling a die: “getting a 3” and “getting a 5”.
    2. Rolling a die: “getting an even number” and “getting an odd number”.
    3. Drawing a card: “getting a heart” and “getting a spade”.
    4. Drawing a card: “getting a red card” and “getting a black card”.
    5. Rolling a die: “getting a number less than 3” and “getting a number greater than 4”.
    6. Rolling a die: “getting a prime number” and “getting an even number”.

  5. Use the complement rule and other probability reasoning to answer each question. Understanding

    1. A spinner has four coloured sections. P(red) = 0.2, P(blue) = 0.35, P(green) = 0.3. Find P(yellow).
    2. Three events A, B, and C are mutually exclusive and exhaustive. P(A) = 1/4 and P(B) = 2/5. Find P(C).
    3. P(winning a game) = 0.6. In 50 games, how many times would you expect to NOT win?
    4. A quality control inspector finds that P(a product is defective) = 0.04. Out of 500 products, how many are expected to be non-defective?

  6. Complementary events problem solving. Problem Solving

    1. A weather forecast says there is a 65% chance of rain tomorrow. What is the probability it will NOT rain? If this forecast is repeated for 20 days with the same probability each day, on how many days would you expect it to be fine (not raining)?
    2. A bag has red and blue balls only. P(red) = 2/5. There are 10 blue balls. How many red balls are in the bag?
    3. Jake says “If P(A) = 0.7 and P(B) = 0.4, then A and B cannot be complementary events.” Is Jake correct? Explain why or why not.
    4. A quiz has 20 true/false questions. A student guesses every answer.
      1. What is the probability of guessing any single question correctly?
      2. What is the probability of getting any single question wrong?
      3. How many questions would you expect the student to get wrong?
  7. Sports raffle. A raffle has 200 tickets. Mia buys 15 tickets. Jordan buys 25 tickets.
    1. Find P(Mia wins).
    2. Find P(Mia does not win) using the complement rule.
    3. Find P(neither Mia nor Jordan wins). Explain your reasoning.
    4. Is “Mia wins” the complement of “Jordan wins”? Justify your answer.
    Problem Solving
  8. Card game strategy. In a card game you need to draw a card that is either a club or a picture card (Jack, Queen, King) from a standard 52-card deck. There are 13 clubs and 12 picture cards; 3 of the picture cards are clubs.
    1. How many cards satisfy the condition (club OR picture card)? Remember not to count any card twice.
    2. Find P(club or picture card).
    3. Use the complement to find P(not a club and not a picture card).
    Problem Solving
  9. Unknown bag. A bag contains only red and blue marbles. When one marble is drawn at random, P(blue) is three times P(red).
    1. Use the complement rule to write an equation and find P(red) and P(blue).
    2. If there are 24 marbles in the bag, how many are red and how many are blue?
    3. Six red marbles are added. Find the new P(red).
    Problem Solving
  10. Medical test. A rapid test for a virus correctly identifies an infected person 92% of the time (P(positive test | infected) = 0.92). For a randomly selected infected person:
    1. What is P(the test gives a negative result) for an infected person? What is this type of result called in medical testing?
    2. Out of 500 infected people tested, how many would you expect to receive an incorrect (negative) result?
    3. Why is it important to understand complementary probabilities in medical testing?
    Problem Solving
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