Let's analyze the problem step by step based on the image provided:

Problem Summary:

1. There are three boxes with cards labeled as follows:

   • Box A contains two cards: one numbered (6) and one labeled with a letter (H).
   • Box P contains four cards labeled as 1, 8, F, and K.
   • Box Q contains five cards labeled as 5, 9, L, M, and N.

2. A card is picked at random from Box A.

   • If the number card 6 is chosen, another card is picked from Box P.
   • If the letter card H is chosen, another card is picked from Box Q.

Task:

We are asked to find the probability that both cards picked are labeled with letters.

Solution Strategy:

1. Step 1: Identify the outcomes where both cards are labeled with letters.

   • If card H is picked from Box A, then the second card is chosen from Box Q, which has the cards: L, M, N.

     • All these cards are labeled with letters.
     • So, the outcomes from this scenario are: (H, L), (H, M), (H, N).

   • If card 6 is picked from Box A, then the second card is chosen from Box P, which has cards: 1, 8, F, K.

     • Only F and K are labeled with letters.
     • So, the outcomes from this scenario are: (6, F), (6, K).

2. Step 2: Calculate the total possible outcomes.

   • From Box A, two cards can be chosen: 6 and H.
   • If 6 is chosen, four cards are possible from Box P (1, 8, F, K).
   • If H is chosen, five cards are possible from Box Q (5, 9, L, M, N).

   So, the total number of possible outcomes is: $2 \times (4 + 5) = 2 \times 9 = 18$

3. Step 3: Calculate the favorable outcomes (both cards are labeled with letters).

   • From the scenario where H is chosen: (H, L), (H, M), (H, N) — 3 favorable outcomes.
   • From the scenario where 6 is chosen: (6, F), (6, K) — 2 favorable outcomes.

   The total number of favorable outcomes is: $3 + 2 = 5$

4. Step 4: Calculate the probability. $\text{Probability} = \frac{\text{favorable outcomes}}{\text{total outcomes}} = \frac{5}{18}$

Final Answer:

The probability that both cards picked are labeled with letters is $\frac{5}{18}$.

Do you have any questions or would you like a detailed breakdown of any step?

Related Questions:

1. What is the probability of drawing a card numbered with digits (instead of letters)?
2. What is the probability that the second card is drawn from Box P?
3. How many outcomes involve drawing card H?
4. What is the probability that a card from Box Q is drawn?
5. If the first card is always picked from Box A, what are the chances of drawing an even-numbered card?

Tip:

When calculating probabilities, always ensure that your set of favorable outcomes is correctly counted and represents all the possibilities for the given event.