The problem describes a class of 32 students. We are given information about how many students study German, Spanish, or neither language. We need to complete a Venn diagram, find the probability that a randomly chosen student studies Spanish but not German, and find the probability that a student who studies German also studies Spanish.

1. Problem Description

The problem describes a class of 32 students. We are given information about how many students study German, Spanish, or neither language. We need to complete a Venn diagram, find the probability that a randomly chosen student studies Spanish but not German, and find the probability that a student who studies German also studies Spanish.

2. Solution Steps

(i) Completing the Venn diagram:

- Total students = 32

- Students studying German = 15

- Students studying Spanish = 18

- Students studying neither = 5

Let $x$ be the number of students studying both German and Spanish.

Students studying only German = 15 - $x$

Students studying only Spanish = 18 - $x$

The sum of all students should be

3

2. $(15 - x) + (18 - x) + x + 5 = 32$

$38 - x = 32$

$x = 38 - 32$

$x = 6$

So, 6 students study both German and Spanish.

Students studying only German = $15 - 6 = 9$

Students studying only Spanish = $18 - 6 = 12$

The Venn diagram should show 9 students studying only German, 12 students studying only Spanish, 6 students studying both, and 5 students studying neither. (Note: the provided image already contained some incorrect numbers within the diagram.)

(ii) Probability of studying Spanish but not German:

The number of students who study Spanish but not German is

1

2. The total number of students is

3

2. The probability is $\frac{12}{32}$.

(iii) Probability of studying Spanish given that the student studies German:

The number of students studying German is

1

5. The number of students studying both German and Spanish is

6. The probability is $\frac{6}{15}$.

3. Final Answer

(i) Venn diagram values:

Only German: 9

Only Spanish: 12

Both: 6

Neither: 5

(ii) Probability of studying Spanish but not German: $\frac{12}{32}$

(iii) Probability of studying Spanish given German: $\frac{6}{15}$