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The Venn diagram problem provides the following information:

- The total number of students is 58.
- 18 students passed only French.
- 11 students passed only German.
- 17 students passed both French and German.
- \( b \) represents the number of students who did not pass either French or German.

### (a) Find the value of \( b \)
To find \( b \), we need to calculate the total number of students who passed either French or German (or both). The total number of students is 58, so the equation will be:

\[
\text{Total students} = (\text{only French}) + (\text{only German}) + (\text{both}) + b
\]

This becomes:

\[
58 = 18 + 11 + 17 + b
\]

Solving for \( b \):

\[
58 = 46 + b \implies b = 58 - 46 = 12
\]

So, the value of \( b \) is 12.

### (b) Probability questions:

#### (i) The probability that a student passed both French and German
The number of students who passed both French and German is 17. The total number of students is 58, so the probability is:

\[
P(\text{both}) = \frac{17}{58}
\]

#### (ii) The probability that a student passed exactly one of these two subjects
The number of students who passed only French or only German is \( 18 + 11 = 29 \). Thus, the probability is:

\[
P(\text{exactly one}) = \frac{29}{58}
\]

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Let me know if you need further details or have questions. Here are 5 follow-up questions you can consider:

1. How would the probabilities change if 5 more students passed only French?
2. What is the probability that a student did not pass either French or German?
3. How would you calculate the probability that a student passed at least one of the subjects?
4. If the number of students who passed both subjects increased by 5, how would that affect \( b \)?
5. What is the probability that a student passed exactly one subject, given they passed at least one?

**Tip**: When dealing with probabilities involving sets, always check for overlaps (like students passing both subjects) to avoid double counting.

The Venn diagram shows the number of students who passed their examination in French (F) and those who passed their examination in German (G). The number of students who did not pass either examination is b. Find the value of b if the total number of students is 58. One of the 58 students is selected at random. Written as a fraction, what is the probability that this student (i) passed both French and German, (ii) passed exactly one of these two subjects?

Solve a Venn diagram probability problem involving students passing exams in French and German. Find the value of b given the total number of students is 58, and calculate the probability that a student passed both or exactly one subject. Suitable for Grades 7-9.

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