Probability Tree Diagrams (B) Worksheet | Fun and Engaging 7th ... - Free Printable

Here are the solutions to the problems on the worksheet, broken down section by section.

Section A: French and German Students

Step 1: Find the total number of students studying German.
We know that $\frac{4}{9}$ of the German students are in Grade 10. The tree diagram shows there are 28 Grade 10 students studying German.
If $\frac{4}{9}$ of the total is 28, we can find the whole group (Total German) like this:
$$28 \div 4 = 7$$ (This is $\frac{1}{9}$ of the group)
$$7 \times 9 = 63$$
So, the total number of students studying German is 63.

Step 2: Fill in the rest of the German column.
* Total German = 63
* Grade 10 German = 28
* Grade 11 German = $63 - 28 =$ 35

Step 3: Find the number of Grade 10 students studying French.
The table says the total number of Grade 10 students is 64.
We already found that 28 of them study German.
* Grade 10 French = $64 - 28 =$ 36

Step 4: Fill in the Tree Diagram for French.
The tree diagram shows a total of 79 students studying French.
* We know Grade 10 French is 36.
* So, Grade 11 French = $79 - 36 =$ 43

Step 5: Complete the Table Totals.
* Grade 11 Row:
* French: 43
* German: 35
* Total: $43 + 35 =$ 78
* French Column Total: Given as 79.
* German Column Total: Calculated as 63.
* Grand Total: $79 + 63 =$ 142 (or $64 + 78 = 142$).

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Section B: Frequency Trees

Problem 1: Sports Center
* Total People: 480
* Members vs Non-members:
* Members are 60% of 480. ($0.60 \times 480 = 288$).
* Non-members are the rest. ($480 - 288 = 192$).
* Inside Members (288 people):
* Pool: $\frac{1}{3}$ of members. ($\frac{1}{3} \times 288 = 96$).
* Gym: The rest. ($288 - 96 = 192$).
* Inside Non-members (192 people):
* Gym: The problem states "75% of all the people... are in the gym."
* Total in Gym = $0.75 \times 480 = 360$.
* We already have 192 members in the gym.
* So, Non-members in Gym = $360 - 192 = 168$.
* Pool: The rest of the non-members. ($192 - 168 = 24$).

Values for Tree 1:
* First split: 288 (Members), 192 (Non-members)
* Members split: 192 (Gym), 96 (Pool)
* Non-members split: 168 (Gym), 24 (Pool)

Problem 2: Supermarket
* Total Customers: 960
* Self-service vs Cashier:
* Ratio is 3:5. Total parts = $3 + 5 = 8$.
* One part = $960 \div 8 = 120$.
* Self-service (3 parts) = $3 \times 120 =$ 360.
* Cashier (5 parts) = $5 \times 120 =$ 600.
* Payment Methods:
* Cashier Branch (600 people):
* Problem says "$\frac{1}{10}$ of all customers pay with cash."
* Total Cash = $\frac{1}{10} \times 960 = 96$.
* We need to figure out how much of that 96 is at the cashier vs self-service. Let's look at the Self-service branch first.
* Self-service Branch (360 people):
* Ratio of Cash to Card is 1:5. Total parts = 6.
* One part = $360 \div 6 = 60$.
* Self-service Cash (1 part) = 60.
* Self-service Card (5 parts) = $5 \times 60 =$ 300.
* Back to Cashier Branch:
* We know Total Cash is 96.
* Self-service Cash is 60.
* So, Cashier Cash = $96 - 60 =$ 36.
* Cashier Card = Total Cashier (600) - Cashier Cash (36) = 564.

Values for Tree 2:
* First split: 360 (Self-service), 600 (Cashier)
* Self-service split: 60 (Cash), 300 (Card)
* Cashier split: 36 (Cash), 564 (Card)

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Section C: School Group Probability

Step 1: Determine the total number of people.
Let $T$ be the total number of people in the group.
* Teachers = $\frac{1}{6} T$
* Students = $\frac{5}{6} T$

Step 2: Analyze the food choices.
* Teachers:
* We are told 2 teachers choose pasta.
* Let's find out how many teachers chose pizza. We don't know yet, so let's look at the students.
* Students:
* 60% of students choose pizza. This means 40% of students choose pasta.
* Number of students choosing pasta = $0.40 \times (\frac{5}{6} T) = \frac{2}{5} \times \frac{5}{6} T = \frac{2}{6} T = \frac{1}{3} T$.
* Total Pizza Eaters:
* The problem says 60% of ALL the group choose pizza.
* Total Pizza = $0.60 T = \frac{3}{5} T$.

Step 3: Solve for T (Total People).
Total Pizza = (Teachers eating Pizza) + (Students eating Pizza)
We know Total Pizza is $\frac{3}{5} T$.
We know Students eating Pizza is 60% of the student population ($\frac{5}{6} T$).
Students eating Pizza = $0.6 \times \frac{5}{6} T = \frac{3}{5} \times \frac{5}{6} T = \frac{3}{6} T = \frac{1}{2} T$.

So, Teachers eating Pizza = Total Pizza - Students eating Pizza
Teachers eating Pizza = $\frac{3}{5} T - \frac{1}{2} T$
To subtract, find a common denominator (10):
$\frac{6}{10} T - \frac{5}{10} T = \frac{1}{10} T$.

So, $\frac{1}{10}$ of the total group are teachers who eat pizza.
We also know that the remaining teachers eat pasta.
Fraction of teachers = $\frac{1}{6} T$.
Teachers eating Pasta = (Total Teachers) - (Teachers eating Pizza)
Teachers eating Pasta = $\frac{1}{6} T - \frac{1}{10} T$
Common denominator is 30:
$\frac{5}{30} T - \frac{3}{30} T = \frac{2}{30} T = \frac{1}{15} T$.

The problem states that 2 teachers choose to eat pasta.
So, $\frac{1}{15} T = 2$.
$T = 2 \times 15 = 30$.
There are 30 people in the group.

Step 4: Calculate the specific numbers for the probability.
* Total People = 30.
* Teachers: $\frac{1}{6}$ of 30 = 5 teachers.
* 2 eat pasta (given).
* 3 eat pizza ($5-2$).
* Students: $\frac{5}{6}$ of 30 = 25 students.
* 60% eat pizza: $0.60 \times 25 = 15$ students eat pizza.
* The rest eat pasta: $25 - 15 = 10$ students eat pasta.

Step 5: Answer the final question.
"What is the probability they chose to eat pasta?"
* Total people eating pasta = (Teachers eating pasta) + (Students eating pasta)
* Total pasta = $2 + 10 = 12$.
* Total people = 30.
* Probability = $\frac{12}{30}$.

Simplify the fraction by dividing top and bottom by 6:
$\frac{12 \div 6}{30 \div 6} = \frac{2}{5}$.

Final Answer:
Section A Table:
* Grade 10: French 36, German 28, Total 64
* Grade 11: French 43, German 35, Total 78
* Total: French 79, German 63, Total 142

Section A Tree Diagram Circles (Left to Right, Top to Bottom):
* First circle (Total): 142
* German branch circle: 63
* French Grade 10: 36
* French Grade 11: 43
* German Grade 11: 35

Section B1 (Sports Center) Tree Values:
* Members: 288, Non-members: 192
* Members -> Gym: 192, Pool: 96
* Non-members -> Gym: 168, Pool: 24

Section B2 (Supermarket) Tree Values:
* Self-service: 360, Cashier: 600
* Self-service -> Cash: 60, Card: 300
* Cashier -> Cash: 36, Card: 564

Section C Answer:
The probability they chose to eat pasta is 2/5 (or 0.4).