Math worksheet for calculating probabilities from a spinner with numbers 4, 5, 6, 10, and 11.

Explanation:
We are given a spinner divided into 6 equal sections. Each section has a number:
- 4
- 5
- 11
- 5
- 6
- 10

Since the spinner is fair and divided equally, each section has the same chance of being landed on. So the probability of landing on any one section is:

\[
\frac{1}{6}
\]

But some numbers appear more than once. Let’s count how many times each number appears:

- 4 → appears 1 time
- 5 → appears 2 times
- 6 → appears 1 time
- 10 → appears 1 time
- 11 → appears 1 time

Total sections = 6 ✔

Now compute each requested probability.

---

1. P(5)
Number 5 appears 2 times out of 6:
\[
P(5) = \frac{2}{6} = \frac{1}{3}
\]

2. P(11)
11 appears 1 time:
\[
P(11) = \frac{1}{6}
\]

3. P(>9)
Numbers greater than 9: 10 and 11
→ 10 appears once, 11 appears once → total 2 favorable outcomes
\[
P(>9) = \frac{2}{6} = \frac{1}{3}
\]

4. P(≤7)
Numbers ≤ 7: 4, 5, 5, 6
→ 4 (1), 5 (2), 6 (1) = 4 outcomes
\[
P(\le 7) = \frac{4}{6} = \frac{2}{3}
\]

5. P(≥12)
Any number ≥12? The largest is 11 → none
\[
P(\ge 12) = \frac{0}{6} = 0
\]

6. P(≤10)
Numbers ≤10: 4, 5, 5, 6, 10 → that’s 5 outcomes
(Only 11 is >10)
\[
P(\le 10) = \frac{5}{6}
\]

7. P(≤3)
Numbers ≤3? None — smallest is 4
\[
P(\le 3) = 0
\]

8. P(≤11)
All numbers are ≤11 (max is 11), so all 6 outcomes
\[
P(\le 11) = \frac{6}{6} = 1
\]

9. P(≤10) — already done above: \( \frac{5}{6} \)

Wait — let’s double-check the list of questions. The worksheet shows two columns:

Left column:
- P(5)
- P(11)
- P(>9)
- P(≤7)
- P(≥12)
- P(≤10)
- P(≤3)

Right column:
- P(≤11)
- P(≤10) ← duplicate? But maybe it's intentional; we’ll compute both as listed.
- P(<12)
- P(≥12) ← duplicate again
- P(≥1)
- P(≥1) ← duplicate? Wait, let’s read carefully.

Actually, from the image description (as given in user input), the right column is:

P(≤11)
P(≤10)
P(<12)
P(≥12)
P(≥1)
P(≤6)

Let me re-parse the original list correctly (based on the text provided):

```
P(5) = P(≤11) =
P(11) = P(≤10) =
P(>9) = P(≥1) =
P(≤7) = P(<12) =
P(≥12) = P(≥12) =
P(≤10) = P(≥1) = ← wait, this seems off — but likely a formatting issue.
P(≤3) = P(≤6) =
```

But looking at standard layout, it's probably:

Left column (7 items):
1. P(5)
2. P(11)
3. P(>9)
4. P(≤7)
5. P(≥12)
6. P(≤10)
7. P(≤3)

Right column (7 items):
1. P(≤11)
2. P(≤10)
3. P(≥1)
4. P(<12)
5. P(≥12)
6. P(≥1)
7. P(≤6)

But duplicates like P(≤10) and P(≥1) appear twice — possibly a typo in transcription, but to be safe, we’ll answer each *unique* expression as it appears, and if repeated, give same answer.

Let’s compute all unique ones:

- P(5) = 2/6 = 1/3
- P(11) = 1/6
- P(>9) = numbers 10, 11 → 2/6 = 1/3
- P(≤7) = 4,5,5,6 → 4/6 = 2/3
- P(≥12) = 0
- P(≤10) = all except 11 → 5/6
- P(≤3) = 0
- P(≤11) = all → 1
- P(≥1) = all numbers are ≥1 → 1
- P(<12) = all numbers are <12 (since max is 11) → 1
- P(≤6) = numbers ≤6: 4, 5, 5, 6 → 4 outcomes → 4/6 = 2/3

Let’s verify values again with full list of outcomes: [4, 5, 11, 5, 6, 10]

Sorted: 4, 5, 5, 6, 10, 11

- ≤6: 4,5,5,6 → 4 → 4/6 = 2/3 ✔
- <12: all 6 → 1 ✔
- ≥1: all 6 → 1 ✔
- ≤11: all 6 → 1 ✔
- ≤10: exclude 11 → 5 → 5/6 ✔
- ≥12: none → 0 ✔
- ≤3: none → 0 ✔
- >9: 10, 11 → 2 → 1/3 ✔
- ≤7: 4,5,5,6 → 4 → 2/3 ✔
- P(5): two 5s → 2/6 = 1/3 ✔
- P(11): one → 1/6 ✔

All consistent.

Now write final answers in order as listed in the problem (left column top to bottom, then right column top to bottom):

Left:
1. P(5) = 1/3
2. P(11) = 1/6
3. P(>9) = 1/3
4. P(≤7) = 2/3
5. P(≥12) = 0
6. P(≤10) = 5/6
7. P(≤3) = 0

Right:
8. P(≤11) = 1
9. P(≤10) = 5/6
10. P(≥1) = 1
11. P(<12) = 1
12. P(≥12) = 0
13. P(≥1) = 1
14. P(≤6) = 2/3

But since the instruction says “Calculate the probability of each spin” and lists those blanks, we should output each value in the order shown.

The original text shows:

P(5) =
P(11) =
P(>9) =
P(≤7) =
P(≥12) =
P(≤10) =
P(≤3) =

P(≤11) =
P(≤10) =
P(≥1) =
P(<12) =
P(≥12) =
P(≥1) =
P(≤6) =

So 14 answers total.

We’ll list them as simplified fractions or whole numbers (0, 1, etc.).

Final Answer:
1/3, 1/6, 1/3, 2/3, 0, 5/6, 0, 1, 5/6, 1, 1, 0, 1, 2/3