1st Student:   P (0.70)                     F (0.30)
/      \                   /      \
2nd Student:  P (0.70)  F (0.30)       P (0.70)  F (0.30)
/   \      /   \         /   \      /   \
3rd Student: P (0.70) F (0.30)  P (0.70) F (0.30)  P (0.70) F (0.30)  P (0.70) F (0.30)

The scenario you provided seems to describe a probability tree, where each student can either pass (P) or fail (F) an exam, with associated probabilities. Let's break down the problem step by step:

1. **1st Student:**
   - Probability of passing (P): 0.70
   - Probability of failing (F): 0.30

2. **2nd Student:**  
   - Given that the first student passes or fails, the second student has:
   - Probability of passing (P): 0.70
   - Probability of failing (F): 0.30

   This pattern continues whether the first student passed or failed.

3. **3rd Student:**
   - Similarly, given the outcomes of the first and second students, the third student also has:
   - Probability of passing (P): 0.70
   - Probability of failing (F): 0.30

Given this structure, the problem likely involves calculating the probability of a particular outcome involving all three students. To calculate this, you multiply the probabilities along the branches of the tree that correspond to the desired outcome.

### Examples of Outcomes and Their Probabilities:
1. **All three students pass (PPP):**
   - Probability = 0.70 × 0.70 × 0.70 = 0.343

2. **First student passes, second fails, third passes (PFP):**
   - Probability = 0.70 × 0.30 × 0.70 = 0.147

3. **All three students fail (FFF):**
   - Probability = 0.30 × 0.30 × 0.30 = 0.027

### If you want to calculate the probability of a specific outcome involving these students, you simply identify the path in the tree and multiply the probabilities along that path.

Would you like to explore any specific outcomes or need further details?

### Relative Questions:
1. What is the probability that exactly two students will pass the exam?
2. How would the probabilities change if the first student's probability of passing was 0.80 instead?
3. What is the probability that at least one student will pass?
4. Can you calculate the probability that exactly one student will fail?
5. How would you determine the expected number of students passing the exam?

**Tip:** When dealing with probability trees, carefully multiplying the probabilities along each path gives you the likelihood of each specific outcome.

Explore how to calculate probabilities using a probability tree with student exam outcomes. This problem involves understanding the likelihood of students passing or failing an exam based on given probabilities. Suitable for advanced high school students and those preparing for probability theory.

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