Mutually Exclusive Events - GCSE Maths - Steps & Examples - Free Printable

Let's solve each section of the worksheet step by step and explain the reasoning.

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Section A: Probability of an event not happening

We use the rule:
> $ P'(A) = 1 - P(A) $

This is because the probability of an event not happening is 1 minus the probability of it happening.

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1) $ P(A) = \frac{3}{4} $
$ P'(A) = 1 - \frac{3}{4} = \frac{1}{4} $

✔ Answer: $ \boxed{\frac{1}{4}} $

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2) $ P(B) = 0.69 $
$ P'(B) = 1 - 0.69 = 0.31 $

✔ Answer: $ \boxed{0.31} $

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3) $ P(C) = \frac{2}{7} $
$ P'(C) = 1 - \frac{2}{7} = \frac{5}{7} $

✔ Answer: $ \boxed{\frac{5}{7}} $

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4) $ P(D) = 57\% = 0.57 $
$ P'(D) = 1 - 0.57 = 0.43 $

✔ Answer: $ \boxed{0.43} $

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5) $ P(E) = 0.125 $
$ P'(E) = 1 - 0.125 = 0.875 $

✔ Answer: $ \boxed{0.875} $

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✔ Section A Answers:

1. $ P'(A) = \frac{1}{4} $
2. $ P'(B) = 0.31 $
3. $ P'(C) = \frac{5}{7} $
4. $ P'(D) = 0.43 $
5. $ P'(E) = 0.875 $

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Section B: Mutually Exclusive Events

Two events are mutually exclusive if they cannot happen at the same time (i.e., no outcome satisfies both).

The spinner has numbers: 1, 2, 3, 4, 5, 6, 7, 8

Let’s evaluate each:

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1) Landing on 2 and 8
- Can you land on both 2 and 8 at the same time? No — only one number comes up.
- So these two outcomes are disjoint.

✔ Yes, mutually exclusive.

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2) Landing on a number less than 4 and greater than 4
- Numbers less than 4: {1, 2, 3}
- Numbers greater than 4: {5, 6, 7, 8}
- Is there any number that is both less than 4 and greater than 4? No.
- Also, note: 4 itself is excluded from both.

✔ Yes, mutually exclusive.

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3) Landing on a factor of 8 and an odd number
- Factors of 8: {1, 2, 4, 8}
- Odd numbers: {1, 3, 5, 7}
- Common outcome? Yes → 1 is both a factor of 8 and odd.

So, it's possible to land on 1 and satisfy both conditions.

✘ Not mutually exclusive (they can happen together).

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4) Landing on 5 and a prime number
- "Landing on 5" is one specific outcome.
- Prime numbers on spinner: {2, 3, 5, 7}
- So landing on 5 is a prime number.

Thus, the event "landing on 5" is included in "landing on a prime number".

So, the two events overlap — 5 is common.

But wait: Are we comparing two events?

Let’s interpret:
- Event 1: Landing on 5
- Event 2: Landing on a prime number

Can both happen at once? Yes — if you land on 5, you’re landing on a prime number.

So they can happen simultaneously.

✘ Not mutually exclusive

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✔ Section B Answers:

1. ✔ Yes
2. ✔ Yes
3. ✘ No
4. ✘ No

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Section C: Exhaustive Events

Events are exhaustive if all possible outcomes are covered (i.e., every result fits at least one event).

Spinner outcomes: {1, 2, 3, 4, 5, 6, 7, 8}

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1) Landing on 2 and 8
- Only two outcomes: 2 and 8
- But other outcomes (like 1, 3, etc.) are not covered.

✘ Not exhaustive

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2) Landing on a number less than 4 and greater than 4
- Less than 4: {1, 2, 3}
- Greater than 4: {5, 6, 7, 8}
- What about 4? It’s neither less than nor greater than 4 → not covered.

✘ Not exhaustive

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3) Landing on a factor of 8 or an odd number
- Factors of 8: {1, 2, 4, 8}
- Odd numbers: {1, 3, 5, 7}
- Union (combined): {1, 2, 3, 4, 5, 7, 8}
- Missing: 6

So 6 is not a factor of 8 and not odd → not covered.

✘ Not exhaustive

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4) Landing on 5 and a prime number
- This is a pair of events:
- Event 1: landing on 5
- Event 2: landing on a prime number
- But many outcomes are not covered: e.g., 1, 2, 3, 4, 6, 7, 8 (only 5 and primes like 2,3,7 are covered)
- But even then: 1 is not prime and not 5 → not covered
- 6 is not prime and not 5 → not covered

So this pair does not cover all outcomes.

✘ Not exhaustive

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Now, write an example of a pair of exhaustive events based on spinning the spinner once.

We need two events such that every outcome is in at least one of them.

Example:
- Even number and Odd number
- Even: {2, 4, 6, 8}
- Odd: {1, 3, 5, 7}
- Together: All numbers → covers everything

✔ Example: Landing on an even number and landing on an odd number

Another valid example:
- Less than 5 and Greater than or equal to 5
- Less than 5: {1, 2, 3, 4}
- ≥5: {5, 6, 7, 8}
- Covers all

Any such pair works.

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✔ Section C Answers:

1. ✘ No
2. ✘ No
3. ✘ No
4. ✘ No

Example of exhaustive events:
→ Landing on an even number and landing on an odd number

(Or: “Landing on a number ≤ 4” and “Landing on a number > 4”)

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Final Summary:

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#### Section A – Complement Probabilities
| Event | P(Event) | P'(Event) |
|-------|----------|-----------|
| A | ¾ | ¼ |
| B | 0.69 | 0.31 |
| C | ⅖ | ⅝ |
| D | 57% | 43% |
| E | 0.125 | 0.875 |

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#### Section B – Mutually Exclusive?
1. Landing on 2 and 8 → ✔ Yes
2. Less than 4 and greater than 4 → ✔ Yes
3. Factor of 8 and odd → ✘ No (1 is both)
4. Landing on 5 and a prime → ✘ No (5 is prime, so overlaps)

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#### Section C – Exhaustive?
1. Landing on 2 and 8 → ✘ No
2. Less than 4 and greater than 4 → ✘ No (4 missing)
3. Factor of 8 or odd → ✘ No (6 missing)
4. Landing on 5 and prime → ✘ No

Example of exhaustive events:
→ Landing on an even number and landing on an odd number

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✔ Final Note: The key idea is:
> If events are mutually exclusive and exhaustive, their probabilities sum to 1.

For example: Even and odd outcomes are mutually exclusive (can't be both), and exhaustive (cover all possibilities). So:
- P(even) + P(odd) = 4/8 + 4/8 = 1

That’s why the title says "Sums to One".

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