To solve the probability question based on the frequency tree provided in the image, let's break it down systematically:

Question:

We are to calculate the probability that a randomly chosen student failed at least one test (either Maths or English).

Frequency Tree Breakdown:

1. Total students who passed Maths:

   • Passed English: 12
   • Failed English: 14
   • Total: $12 + 14 = 26$

2. Total students who failed Maths:

   • Passed English: 15
   • Failed English: 24
   • Total: $15 + 24 = 39$

3. Total number of students:

   • $26 + 39 = 65$

What does "failed at least one test" mean?

• A student fails at least one test if they:
  • Fail Maths, or
  • Fail English, or
  • Fail both.

The complement of this is passing both tests, so we can calculate this using:

$P(\text{failed at least one test}) = 1 - P(\text{passed both tests})$

Probability of passing both tests:

From the frequency tree:

• Students who passed both Maths and English: $12$
• Total students: $65$

$P(\text{passed both tests}) = \frac{12}{65}$

Probability of failing at least one test:

$P(\text{failed at least one test}) = 1 - \frac{12}{65} = \frac{65}{65} - \frac{12}{65} = \frac{53}{65}$

Final Answer:

The probability that a randomly chosen student failed at least one test is:

$\boxed{\frac{53}{65}}$

Let me know if you'd like me to explain this in further detail or have any questions!

Here are 5 related questions to deepen understanding:

1. What is the probability of a student failing both tests?
2. How many students failed only English?
3. How many students passed exactly one test?
4. What is the probability of a student passing only Maths?
5. If another student is added to the group and passes both tests, how does the probability of failing at least one test change?

Tip: Always consider the complement rule when dealing with probabilities involving "at least" or "at most"!