Let's work through the problem step by step.

(a) Theoretical Probability

To find the theoretical probability of both choosing the 2 or 3 ball and drawing the Q card:

1. Total possible outcomes:
   There are 3 balls ($1, 2, 3$) and 3 cards ($K, Q, J$), so the total number of outcomes is: $3 \times 3 = 9$

2. Favorable outcomes:

   • Choosing $2Q$ or $3Q$ means two outcomes are favorable: $2Q$ and $3Q$.

Thus, the probability of this event is: $\text{Theoretical Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{2}{9}$

Rounding to the nearest thousandth: $\frac{2}{9} \approx 0.222$

(b) Experimental Probability

To find the experimental probability, we use the observed data:

1. Number of trials with $2Q$ or $3Q$:
   From the data:

   • $2Q$ occurred 62 times.
   • $3Q$ occurred 60 times.

   Total favorable outcomes: $62 + 60 = 122$

2. Total number of trials:
   The total number of trials is given as 550.

Thus, the experimental probability is: $\text{Experimental Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total trials}} = \frac{122}{550}$

Simplify and round to the nearest thousandth: $\frac{122}{550} \approx 0.222$

(c) Choosing the Correct Statement

The correct statement is:

• "With a large number of trials, there might be a difference between the experimental and theoretical probabilities, but the difference should be small."

This aligns with the Law of Large Numbers, which states that as the number of trials increases, the experimental probability will approach the theoretical probability.

Final Answers:

• (a) Theoretical Probability: $0.222$
• (b) Experimental Probability: $0.222$
• (c) Correct statement:
  "With a large number of trials, there might be a difference between the experimental and theoretical probabilities, but the difference should be small."

Would you like further clarification or additional assistance?

Related Questions:

1. How would the theoretical probability change if the number of balls or cards increased?
2. What are the main factors that could cause the experimental probability to deviate from the theoretical probability?
3. How does the Law of Large Numbers apply to experimental probability?
4. How can we visually compare theoretical and experimental probabilities?
5. What are common errors to avoid when calculating probabilities?

Tip:

When solving probability problems, always verify whether you're calculating theoretical or experimental probabilities to avoid confusion.