Middle School Probability Worksheets - Cazoom Math Worksheets - Free Printable

Let’s go step by step through each section of the worksheet.

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Section A: Probability of an event not happening

We use this rule:
If P(A) is the probability that event A happens, then
P(not A) = 1 – P(A)

Also remember:
- Fractions stay as fractions unless told otherwise.
- Decimals and percentages can be converted to decimals for calculation (e.g., 57% = 0.57).

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Problem 1:
Given: P(A) = 3/4
→ P(not A) = 1 – 3/4 = 1/4

Given: P(C) = 2/7
→ P(not C) = 1 – 2/7 = 5/7

Given: P(E) = 0.125
→ P(not E) = 1 – 0.125 = 0.875

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Problem 2:
Given: P(B) = 0.69
→ P(not B) = 1 – 0.69 = 0.31

Given: P(D) = 57% → convert to decimal: 0.57
→ P(not D) = 1 – 0.57 = 0.43 or 43%

*(Note: Since input was in percent, output can also be in percent — but either is acceptable unless specified. We’ll write both forms where helpful.)*

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Section B: Mutually Exclusive Events

Mutually exclusive means: two events cannot happen at the same time.

Spinner has numbers: 1, 2, 3, 4, 5, 6, 7, 8 (from image description — standard spinner with 8 sections labeled 1–8)

Check each pair:

1) Landing on 2 and 8 → Can you land on both at once? No → Mutually exclusive ✔

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}
No overlap → Mutually exclusive ✔

Wait — what about 4 itself? The condition says “less than 4” and “greater than 4”, so 4 is excluded from both. So no common outcome → still mutually exclusive.

But let’s double-check: Is there any number that is BOTH <4 and >4? No → ✔

3) Landing on a factor of 8 and an odd number
Factors of 8: 1, 2, 4, 8
Odd numbers on spinner: 1, 3, 5, 7
Common number: 1 → You CAN land on 1, which is both a factor of 8 AND odd → NOT mutually exclusive ✘

4) Landing on 5 and a prime number
Prime numbers between 1–8: 2, 3, 5, 7
So 5 IS a prime number → If you land on 5, you’ve landed on a prime number too → These events CAN happen together → NOT mutually exclusive ✘

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Section C: Exhaustive Events

Exhaustive means: together, they cover ALL possible outcomes.

Again, spinner: 1 to 8.

1) Landing on 2 and 8 → Only covers two outcomes. What about 1,3,4,5,6,7? Not covered → Not exhaustive ✘

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 missing! → Does NOT cover all outcomes → Not exhaustive ✘

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: {1,2,3,4,5,7,8} → Missing 6 → 6 is not a factor of 8 and not odd → So not covered → Not exhaustive ✘

Wait — let me check again:

Is 6 a factor of 8? No. Is 6 odd? No. So yes, 6 is missing.

But wait — maybe I made a mistake? Let’s list all numbers 1–8:

1 → factor of 8? Yes. Odd? Yes → covered
2 → factor? Yes → covered
3 → odd? Yes → covered
4 → factor? Yes → covered
5 → odd? Yes → covered
6 → factor? No. Odd? No → NOT covered
7 → odd? Yes → covered
8 → factor? Yes → covered

Yes, 6 is missing → not exhaustive.

4) Landing on 5 and a prime number
This is tricky wording. “Landing on 5 and a prime number” — does it mean the event is “landing on 5” AND “landing on a prime number”? That would only be true if you land on 5 (since 5 is prime). But as a combined event, it’s just landing on 5.

But the question says: “decide whether the events are exhaustive or not.” Wait — looking back at original:

> For each question decide whether the events are exhaustive or not.

And item 4 says: “Landing on 5 and a prime number”

Actually, re-reading Section C intro:

> The following questions are based on the same spinner being spun once. For each question decide whether the events are exhaustive or not.

Each item lists TWO events? Or one compound event?

Looking at structure:

In Section B, each item had two conditions joined by “and” — meaning we’re checking if those two specific outcomes can occur together.

In Section C, same format — but now asking if the EVENTS listed are exhaustive.

Wait — actually, rereading carefully:

Section C says:

> A group of events are exhaustive if they cover all possible outcomes. The following questions are based on the same spinner being spun once. For each question decide whether the events are exhaustive or not.

Then items:

1) Landing on 2 and 8 → probably means the set {2, 8} — do these two outcomes cover all possibilities? No.

Similarly, 2) Landing on a number less than 4 and greater than 4 → meaning the union of those two sets: {1,2,3} ∪ {5,6,7,8} = {1,2,3,5,6,7,8} — missing 4 → not exhaustive.

3) Landing on a factor of 8 or an odd number → note: here it says “or”, unlike previous “and”. In math, “or” usually means union.

So: factors of 8: {1,2,4,8}, odd numbers: {1,3,5,7} → union: {1,2,3,4,5,7,8} — missing 6 → not exhaustive.

4) Landing on 5 and a prime number → again, “and” — so intersection? Or is it listing two separate events?

Actually, looking at the pattern, in Section C, each item seems to describe ONE composite event or a pair of events whose union we consider.

But item 4 says “Landing on 5 and a prime number” — since 5 is a prime number, this is equivalent to just “landing on 5”. So the event is {5}. Clearly doesn’t cover all outcomes → not exhaustive.

BUT — perhaps the intention is different? Maybe in item 4, it's meant to be two events: “landing on 5” and “landing on a prime number” — and we take their union?

That would make more sense for exhaustiveness.

Let me reinterpret Section C items as describing TWO events, and we are to consider whether the UNION of those two events covers all outcomes.

That matches the definition: “a group of events are exhaustive if they cover all possible outcomes.”

So:

Item 1: Event A = land on 2; Event B = land on 8 → Union = {2,8} → not exhaustive.

Item 2: Event A = land on <4 → {1,2,3}; Event B = land on >4 → {5,6,7,8}; Union = {1,2,3,5,6,7,8} → missing 4 → not exhaustive.

Item 3: Event A = land on factor of 8 → {1,2,4,8}; Event B = land on odd number → {1,3,5,7}; Union = {1,2,3,4,5,7,8} → missing 6 → not exhaustive.

Item 4: Event A = land on 5 → {5}; Event B = land on prime number → {2,3,5,7}; Union = {2,3,5,7} → missing 1,4,6,8 → definitely not exhaustive.

Wait — but 1 is not prime? Correct, 1 is not considered a prime number. Primes start at 2.

So union is {2,3,5,7} — misses 1,4,6,8 → not exhaustive.

But hold on — is there any item that IS exhaustive?

None so far. But let’s think — maybe I misread item 3.

Item 3: “Landing on a factor of 8 or an odd number” — if “or” means union, and we have {1,2,3,4,5,7,8}, missing only 6.

Still not exhaustive.

Perhaps none are exhaustive? That’s possible.

But let’s check the last part:

> Write down an example of a pair of exhaustive events based on spinning this spinner once.

So we need to create our own.

Example: “Landing on an even number or landing on an odd number” → covers all 1–8 → exhaustive.

Or: “Landing on a number ≤4 or >4” → {1,2,3,4} ∪ {5,6,7,8} = all → exhaustive.

Good.

Now, back to answering the given items.

For Section C, all four given pairs are NOT exhaustive.

But let me confirm item 2 again: “Landing on a number less than 4 and greater than 4”

If interpreted as two separate events: A = <4, B = >4, then union excludes 4 → not exhaustive.

If someone misreads “and” as requiring both, that wouldn’t make sense for exhaustiveness — because no single spin can satisfy both.

So definitely, we consider the union of the two described events.

Thus:

C1: {2,8} → not exhaustive
C2: {1,2,3,5,6,7,8} → missing 4 → not exhaustive
C3: {1,2,3,4,5,7,8} → missing 6 → not exhaustive
C4: {5} ∪ {2,3,5,7} = {2,3,5,7} → missing 1,4,6,8 → not exhaustive

All are not exhaustive.

But wait — in item 4, if “landing on 5 and a prime number” is meant to be a single event (i.e., landing on a number that is both 5 and prime), then it’s just {5}, still not exhaustive.

Either way, same conclusion.

Now, final instruction: Write an example of a pair of exhaustive events.

As above: “Landing on an even number or landing on an odd number” — since every number is either even or odd, and 1–8 includes both, this covers all.

Another: “Landing on a number less than or equal to 4 or greater than 4” — covers all.

I’ll pick the first one.

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Now, compiling answers.

Final Answer:

Section A:
1) P(not A) = 1/4; P(not C) = 5/7; P(not E) = 0.875
2) P(not B) = 0.31; P(not D) = 0.43 (or 43%)

Section B:
1) Mutually exclusive
2) Mutually exclusive
3) Not mutually exclusive
4) Not mutually exclusive

Section C:
1) Not exhaustive
2) Not exhaustive
3) Not exhaustive
4) Not exhaustive
Example of exhaustive events: Landing on an even number or landing on an odd number.

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Final Answer:
Section A:
1) P(not A) = \frac{1}{4}, P(not C) = \frac{5}{7}, P(not E) = 0.875
2) P(not B) = 0.31, P(not D) = 0.43

Section B:
1) Mutually exclusive
2) Mutually exclusive
3) Not mutually exclusive
4) Not mutually exclusive

Section C:
1) Not exhaustive
2) Not exhaustive
3) Not exhaustive
4) Not exhaustive
Example: Landing on an even number or landing on an odd number.