https://file.aiquickdraw.com/mathbot/file/1727194372753f0ygjm5a.jpg

Let's break down and solve the probability questions from the image:

The problem presents three boxes (A, P, Q) containing labeled cards. 

- **Box A** has two cards: \(6\) (number) and \(H\) (letter).
- **Box P** has four cards: \(1, 8, F, K\) (two numbers, two letters).
- **Box Q** has five cards: \(5, 9, L, M, N\) (two numbers, three letters).

For the problem:

- **Step 1:** A card is picked at random from **Box A**. 
  - If the **number** \(6\) is chosen, a card is then picked randomly from **Box P**.
  - If the **letter** \(H\) is chosen, a card is picked randomly from **Box Q**.

We are asked to find two probabilities:
1. The probability that both cards are labeled with letters.
2. The probability that one card is labeled with a number, and the other with a letter.

---

### (i) Probability that **both cards are labeled with letters**:
- **Two ways to get letters**:
  - If \(H\) is selected from **Box A**, we pick from **Box Q**. The possible letters from **Box Q** are \(L, M, N\) (3 letters out of 5 total cards).
  - If \(6\) is selected from **Box A**, we pick from **Box P**. The possible letters from **Box P** are \(F, K\) (2 letters out of 4 total cards).

The total probability of selecting letters in both cases is:
- Probability of picking \(H\) from **Box A**: \( \frac{1}{2} \)
  - Then selecting a letter from **Box Q**: \( \frac{3}{5} \)
- Probability of picking \(6\) from **Box A**: \( \frac{1}{2} \)
  - Then selecting a letter from **Box P**: \( \frac{2}{4} = \frac{1}{2} \)

Thus, the total probability is:
\[
P(\text{both letters}) = \left( \frac{1}{2} \times \frac{3}{5} \right) + \left( \frac{1}{2} \times \frac{1}{2} \right)
= \frac{3}{10} + \frac{1}{4}
\]

Finding a common denominator (20):
\[
P(\text{both letters}) = \frac{6}{20} + \frac{5}{20} = \frac{11}{20}
\]

### (ii) Probability that **one card is labeled with a number and the other with a letter**:
This situation has two cases:
1. Picking a **number** first and then a **letter**.
2. Picking a **letter** first and then a **number**.

Let's calculate each case:
- **Case 1**: Picking a number from **Box A** (\(6\)) and then a letter from **Box P**.
  - Probability of picking \(6\): \( \frac{1}{2} \)
  - Probability of picking a letter from **Box P** (\(F, K\)): \( \frac{2}{4} = \frac{1}{2} \)

- **Case 2**: Picking a letter from **Box A** (\(H\)) and then a number from **Box Q**.
  - Probability of picking \(H\): \( \frac{1}{2} \)
  - Probability of picking a number from **Box Q** (\(5, 9\)): \( \frac{2}{5} \)

Thus, the total probability is:
\[
P(\text{number and letter}) = \left( \frac{1}{2} \times \frac{1}{2} \right) + \left( \frac{1}{2} \times \frac{2}{5} \right)
= \frac{1}{4} + \frac{2}{10} = \frac{1}{4} + \frac{1}{5}
\]

Finding a common denominator (20):
\[
P(\text{number and letter}) = \frac{5}{20} + \frac{4}{20} = \frac{9}{20}
\]

---

### Final Answers:
1. **Probability that both cards are labeled with letters**: \( \frac{11}{20} \)
2. **Probability that one card is labeled with a number and the other with a letter**: \( \frac{9}{20} \)

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Let me know if you'd like further clarification or additional details. Here are five related questions:

1. What is the probability of drawing a number from both boxes?
2. How does the probability change if an additional card is added to Box A?
3. What if Box P only contained letters? How would that affect the probabilities?
4. How would the probabilities differ if Box Q only had numbers?
5. What if Box A had three cards (two numbers and one letter)? How would the new probabilities be calculated?

**Tip**: Always break the probability into individual outcomes and add them together to find the total probability.

By listing the outcomes at an event, find the probability that (i) both cards are labeled with letters, (ii) one card is labeled with a number and another card is labeled with a letter.

Solve a probability problem involving labeled cards from different boxes. Learn how to calculate the probability of selecting two letters or a combination of a number and a letter from different boxes. Suitable for Grades 10-12.

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