The probability that a student correctly answers on the first try (the event A) is P(A) = 0.2. If the student answers incorrectly on the first try, the student is allowed a second try to correctly answer the question (the event B). The probability that the student answers correctly on the second try given that he answered incorrectly on the first try is 0.3. Find the probability that the student correctly answers the question on the first or second try.

To solve this problem, we need to determine the probability that a student answers the question correctly on either the first or second try.

### Given:
- \( P(A) = 0.2 \), the probability that the student answers correctly on the first try.
- \( P(B|A^c) = 0.3 \), the probability that the student answers correctly on the second try given that they answered incorrectly on the first try. \( A^c \) denotes the complement of event \( A \), meaning the student answered incorrectly on the first try.

### Step 1: Probability that the student answers incorrectly on the first try
The probability of the student answering incorrectly on the first try is the complement of answering correctly on the first try:
\[
P(A^c) = 1 - P(A) = 1 - 0.2 = 0.8
\]

### Step 2: Probability that the student answers correctly on the second try
The student can only answer correctly on the second try if they answered incorrectly on the first try. The joint probability that the student answers incorrectly on the first try and then correctly on the second try is:
\[
P(A^c \cap B) = P(A^c) \times P(B|A^c) = 0.8 \times 0.3 = 0.24
\]

### Step 3: Total probability of answering correctly on the first or second try
The total probability that the student answers correctly on either the first or second try is the sum of the probabilities of:
1. Answering correctly on the first try (\( P(A) \)).
2. Answering incorrectly on the first try and correctly on the second try (\( P(A^c \cap B) \)).

Thus, the total probability is:
\[
P(\text{Correct on first or second try}) = P(A) + P(A^c \cap B) = 0.2 + 0.24 = 0.44
\]

### Final Answer:
The probability that the student correctly answers the question on the first or second try is \( \boxed{0.44} \).

Let me know if you'd like more details or have any questions!

Here are 5 related questions:
1. What is the probability that the student answers incorrectly on both tries?
2. How would the probability change if the probability of answering correctly on the second try was 0.5?
3. What is the complement of the probability that the student answers correctly on either the first or second try?
4. How do you calculate conditional probability in general?
5. What happens to the total probability if \( P(A) \) increases?

**Tip:** Conditional probability focuses on finding the likelihood of an event happening given that another event has already occurred.

Learn how to calculate the probability that a student answers correctly on either the first or second try with step-by-step guidance. This problem involves key concepts in probability, such as complementary events and conditional probability, and is suitable for students in Grades 9-12.

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