Compound probability worksheet with a frequency table on student movie preferences and related probability questions.

Let’s go step by step through each question using the table.

We are told that 96 students were interviewed total (bottom right corner of table).

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Question 9: How many students were interviewed?
→ Look at the “TOTAL” row and “TOTAL” column → 96

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Question 10: How many freshmen like drama?
→ Find row “Freshmen (F)” and column “Drama (D)” → 3

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Question 11: How many sophomores like romance?
→ Row “Sophomores (S)”, column “Romance (R)” → 30

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Question 12: Find P(D ∪ F)
This means: Probability that a student likes Drama OR is a Freshman (or both).

Formula:
P(D ∪ F) = P(D) + P(F) - P(D ∩ F)

From table:
- Total students = 96
- D (Drama total) = 13 → P(D) = 13/96
- F (Freshmen total) = 48 → P(F) = 48/96
- D F (Freshmen who like Drama) = 3 → P(D ∩ F) = 3/96

So:
P(D ∪ F) = 13/96 + 48/96 - 3/96 = (13 + 48 - 3)/96 = 58/96

Simplify: divide numerator and denominator by 2 → 29/48

*(But we can leave as 58/96 unless asked to simplify — let’s check if simplification is needed. Since other answers may be left unsimplified, we’ll keep it as 58/96 for now, but note that 29/48 is simpler.)*

Actually, let’s double-check with counting:

Number of students who are Freshmen OR like Drama:

- All Freshmen: 48
- Plus Drama lovers who are NOT Freshmen: Sophomores who like Drama = 10
→ So total = 48 + 10 = 58 → Yes! So 58/96

✔ Final: 58/96 or simplified 29/48

We’ll use 58/96 unless specified otherwise — but since probability questions often want simplified fractions, let’s go with 29/48.

Wait — let’s see what the worksheet expects. Looking at later questions, they might expect unreduced. But standard practice is to reduce. Let’s do both in reasoning, pick reduced for final.

Actually, let’s hold off and compute all first.

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Question 13: Find P(A ∪ S)
Probability student likes Action OR is a Sophomore (or both)

A = Action total = 30
S = Sophomores total = 48
A ∩ S = Sophomores who like Action = 6

So:
P(A ∪ S) = P(A) + P(S) - P(A ∩ S) = 30/96 + 48/96 - 6/96 = (30+48-6)/96 = 72/96

Simplify: divide by 24 → 3/4? Wait: 72 ÷ 24 = 3, 96 ÷ 24 = 4 → yes, 3/4

Or divide by 12: 6/8 → then 3/4. Same.

Counting method:

All Sophomores: 48
Plus Action lovers who are NOT Sophomores: Freshmen who like Action = 24
Total = 48 + 24 = 72 → 72/96 = 3/4 ✔

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Question 14: Find P(R ∩ S)
Probability student likes Romance AND is a Sophomore

Look at intersection: Romance column, Sophomore row → 30

So P(R ∩ S) = 30/96

Simplify: divide by 6 → 5/16

But again, maybe leave as 30/96? Let’s see context.

Actually, this is an intersection, not union — so no subtraction needed. Just count the cell.

✔ 30/96 or 5/16

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Question 15: Find P(H ∩ F)
Horror AND Freshman

Cell: Horror column, Freshman row → 18

So P(H ∩ F) = 18/96

Simplify: divide by 6 → 3/16

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Question 16: What is the probability that a student is either a freshman or likes drama (or both)?

This is same as Q12: P(F ∪ D) = 58/96 = 29/48

Already did this.

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Question 17: What is the probability that a student is a sophomore or likes horror (or both)?

P(S ∪ H) = P(S) + P(H) - P(S ∩ H)

S = 48, H = 20, S ∩ H = 2 (sophomores who like horror)

So: 48/96 + 20/96 - 2/96 = (48+20-2)/96 = 66/96

Simplify: divide by 6 → 11/16

Counting: All sophomores (48) + horror lovers who are not sophomores (freshmen who like horror = 18) → 48 + 18 = 66 → yes.

✔ 66/96 or 11/16

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Question 18: What is the probability that a student likes action?

Action total = 30 → P(A) = 30/96 = 5/16 after simplifying (divide by 6)

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Now, let’s compile final answers. We should present them clearly.

Note: For probabilities, unless specified, simplified fraction is best.

But let’s check if any question asks for decimal or percent — no, all seem to want fraction.

Also, for unions, we used formula correctly.

Double-check Q12 and Q16 — same thing, good.

Q13: A ∪ S = 72/96 = 3/4 — correct.

Q14: R ∩ S = 30/96 = 5/16 — correct.

Q15: H ∩ F = 18/96 = 3/16 — correct.

Q17: S ∪ H = 66/96 = 11/16 — correct.

Q18: A = 30/96 = 5/16 — correct.

Now, write final answers in order.

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Final Answer:
9. 96
10. 3
11. 30
12. 29/48
13. 3/4
14. 5/16
15. 3/16
16. 29/48
17. 11/16
18. 5/16