Worksheet titled "Probability Tree Diagrams (B)" from Cazoom Maths Resources, designed to teach students how to complete frequency trees and two-way tables using real-world scenarios involving students, sports centers, supermarkets, and school groups.

Let's solve each section of this Probability Tree Diagrams (B) worksheet step by step, with clear explanations.

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Section A: Students Studying French or German

We are given:
- Total number of students studying German = 79
- $\frac{4}{9}$ of the students studying German are in Grade 10
- Grade 10 total = 64 (this includes both French and German students)

#### Step 1: Find how many German students are in Grade 10 and Grade 11

$$
\text{Grade 10 German} = \frac{4}{9} \times 79 = \frac{316}{9} \approx 35.11
$$

Wait — that’s not a whole number! But we’re told there are 28 Grade 10 students studying German from the tree diagram.

Wait — look at the tree diagram:

- The "German" branch leads to:
- Grade 10: 28
- Grade 11: ?

So we can use this to find total German students.

But the total German students is labeled as 79, so:

$$
\text{Grade 11 German} = 79 - 28 = 51
$$

Now, check if $\frac{4}{9}$ of German students are in Grade 10:

$$
\frac{28}{79} \approx 0.354 \quad \text{but} \quad \frac{4}{9} \approx 0.444
$$

Not equal. So contradiction?

Wait — maybe the 79 is not the total German students?

Look again: the tree diagram shows:

- From root → French → Grade 10: blank
- French → Grade 11: blank
- German → Grade 10: 28
- German → Grade 11: blank
- And the node after "German" is 79

Ah! The 79 is the number of students who study German, so total German = 79.

So:

- Grade 10 German = 28
- Grade 11 German = $79 - 28 = 51$

But we're told: $\frac{4}{9}$ of German students are in Grade 10.

Check: $\frac{28}{79} \approx 0.354$, but $\frac{4}{9} \approx 0.444$. Not equal.

So inconsistency?

Wait — perhaps the 79 is not the total German? Let's re-express.

Wait — actually, the circle labeled "79" is on the "French" branch?

No — let's read carefully.

The diagram:

```
[Root]
/ \
French German
| |
79 [?]
| |
Grade 10 Grade 10
| |
[ ] 28
|
Grade 11
|
[ ]
```

Wait — no. Actually:

- Root splits into French and German
- The French branch has a node labeled 79
- Then French splits into Grade 10 and Grade 11
- German splits into Grade 10 (28) and Grade 11 (?)

So:

- Number of students studying French = 79
- Number of students studying German = ? (but we know Grade 10 German = 28)

But wait — the problem says:

> "$\frac{4}{9}$ of the students studying German are in grade 10."

So let $ G $ be total German students.

Then:
$$
\frac{4}{9}G = \text{Grade 10 German} = 28
$$

Solve:
$$
G = 28 \times \frac{9}{4} = 63
$$

So total German students = 63

Then Grade 11 German = $63 - 28 = 35$

Now, total French = 79 (from diagram)

Now, total students in Grade 10 = 64 (given)

So Grade 10 French = ?
Grade 10 German = 28

So:
$$
\text{Grade 10 French} = 64 - 28 = 36
$$

Then Grade 11 French = $79 - 36 = 43$

Grade 11 German = 35 (already found)

Now fill in the two-way table:

| | French | German | Total |
|-----------|--------|--------|-------|
| Grade 10 | 36 | 28 | 64 |
| Grade 11 | 43 | 35 | 78 |
| Total | 79 | 63 | 142 |

Now complete the frequency tree:

- Root → French: 79
- Grade 10: 36
- Grade 11: 43
- Root → German: 63
- Grade 10: 28
- Grade 11: 35

So fill in the blanks:

- First circle (after root): 79 (French), 63 (German)
- French → Grade 10: 36, Grade 11: 43
- German → Grade 10: 28, Grade 11: 35

✔ Done for Section A.

---

Section B

#### 1) Sports Center Problem

Given:
- Total people: 480
- 60% are members → $0.6 \times 480 = 288$ members
- Non-members = $480 - 288 = 192$
- $\frac{1}{3}$ of members are using the pool → $\frac{1}{3} \times 288 = 96$ members in pool
- So members in gym = $288 - 96 = 192$
- 75% of all people are in gym → $0.75 \times 480 = 360$ in gym
- So non-members in gym = $360 - 192 = 168$
- Non-members in pool = $192 - 168 = 24$

Now build frequency tree:

- Root: 480
- Members: 288
- Gym: 192
- Pool: 96
- Non-members: 192
- Gym: 168
- Pool: 24

Fill in:

- Root: 480
- Members: 288
- Non-members: 192
- Members → Gym: 192, Pool: 96
- Non-members → Gym: 168, Pool: 24

✔ Done.

---

#### 2) Supermarket Problem

Given:
- Total customers: 960
- 960 use self-service counters
- Ratio of self-service : cashier = 3:5

Wait — “960 customers use the self-service counters” — that seems like it's already the count.

But then: “The ratio of customers using self-service to cashier is 3:5”

But if 960 use self-service, then:

Let’s suppose:
- Self-service: $3x$
- Cashier: $5x$
- Total = $8x = 960$ → $x = 120$

Then:
- Self-service: $3 \times 120 = 360$
- Cashier: $5 \times 120 = 600$

But the problem says: “960 customers use the self-service counters” — contradiction.

Wait — reread:

> "In one day at a supermarket, 960 customers use the self-service counters."

So self-service = 960

Then “The ratio of customers using self-service counters to those using a cashier is 3:5”

That would mean:
$$
\frac{\text{self-service}}{\text{cashier}} = \frac{3}{5}
\Rightarrow \frac{960}{\text{cashier}} = \frac{3}{5}
\Rightarrow \text{cashier} = \frac{5}{3} \times 960 = 1600
$$

Total customers = $960 + 1600 = 2560$

But that’s fine.

Now:
- At self-service:
- Ratio cash : card = 1:5
- So total parts = 6
- Cash: $ \frac{1}{6} \times 960 = 160 $
- Card: $ \frac{5}{6} \times 960 = 800 $

Also: “$\frac{1}{10}$ of all customers pay with cash”

Total customers = 960 (self) + 1600 (cashier) = 2560

Cash payments: $ \frac{1}{10} \times 2560 = 256 $

We already have 160 cash at self-service, so cashier cash = $256 - 160 = 96$

Then cashier card = $1600 - 96 = 1504$

Now build tree:

- Root: 2560
- Self-service: 960
- Cash: 160
- Card: 800
- Cashier: 1600
- Cash: 96
- Card: 1504

Fill in:

- Root: 2560
- Self-service: 960
- Cashier: 1600
- Self-service → Cash: 160, Card: 800
- Cashier → Cash: 96, Card: 1504

✔ Done.

---

Section C: Restaurant Problem

Given:
- Group of teachers and students go to restaurant.
- 60% choose pizza → 40% choose pasta
- 60% of students choose pizza → 40% choose pasta
- $\frac{1}{6}$ of group are teachers → $\frac{5}{6}$ are students
- 2 teachers choose pasta
- Find: Probability a randomly selected person chose pasta

Let total group size = $ x $

Then:
- Teachers = $ \frac{1}{6}x $
- Students = $ \frac{5}{6}x $

Let’s assume $ x = 6k $ to avoid fractions.

Then:
- Teachers = $ k $
- Students = $ 5k $

Now:
- 60% of group choose pizza → $ 0.6 \times 6k = 3.6k $ choose pizza
- So pasta = $ 6k - 3.6k = 2.4k $

We are told: 2 teachers choose pasta

So:
- Teachers choosing pasta = 2
- Teachers choosing pizza = $ k - 2 $

Now, 60% of students choose pizza → $ 0.6 \times 5k = 3k $ students choose pizza
- So students choosing pasta = $ 5k - 3k = 2k $

Now total pasta eaters = teachers (pasta) + students (pasta) = $ 2 + 2k $

But earlier we said total pasta eaters = $ 2.4k $

So:
$$
2 + 2k = 2.4k \Rightarrow 2 = 0.4k \Rightarrow k = 5
$$

Now compute:

- $ k = 5 $
- Total group = $ 6k = 30 $
- Teachers = 5
- Students = 25
- Teachers choosing pasta = 2
- Students choosing pasta = $ 2k = 10 $
- Total pasta eaters = $ 2 + 10 = 12 $

So probability a random person chose pasta = $ \frac{12}{30} = \boxed{\frac{2}{5}} $

Now complete the frequency tree:

- Root: 30
- Teacher: 5
- Pizza: $5 - 2 = 3$
- Pasta: 2
- Student: 25
- Pizza: $0.6 \times 25 = 15$
- Pasta: $25 - 15 = 10$

So fill in:

- Root: 30
- Teacher: 5
- Student: 25
- Teacher → Pizza: 3, Pasta: 2
- Student → Pizza: 15, Pasta: 10

Answer: Probability they chose pasta = $ \boxed{\frac{2}{5}} $

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✔ Final Answers Summary

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Section A

Two-way Table:

| | French | German | Total |
|-----------|--------|--------|-------|
| Grade 10 | 36 | 28 | 64 |
| Grade 11 | 43 | 35 | 78 |
| Total | 79 | 63 | 142 |

Tree Diagram:
- French: 79 → Grade 10: 36, Grade 11: 43
- German: 63 → Grade 10: 28, Grade 11: 35

---

Section B

#### 1) Sports Center

- Root: 480
- Members: 288 → Gym: 192, Pool: 96
- Non-members: 192 → Gym: 168, Pool: 24

#### 2) Supermarket

- Root: 2560
- Self-service: 960 → Cash: 160, Card: 800
- Cashier: 1600 → Cash: 96, Card: 1504

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Section C

- Root: 30
- Teacher: 5 → Pizza: 3, Pasta: 2
- Student: 25 → Pizza: 15, Pasta: 10

Answer: Probability of choosing pasta = $ \boxed{\frac{2}{5}} $

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Let me know if you'd like a visual version of any tree!