It seems you have uploaded an image of a problem related to experimental and theoretical probability. Let's break it down:

Part (a) - Theoretical Probability:

You are tasked with finding the theoretical probability of drawing a ball that is odd and a card that is either a K or a J in a single trial.

• The balls are numbered 1, 2, and 3, so the odd-numbered balls are 1 and 3.
• The cards are labeled K, Q, and J, so the cards that are either a K or a J are K and J.

The theoretical probability is calculated by multiplying the probability of drawing an odd ball and the probability of drawing a K or J card:

• Probability of drawing an odd ball = 2/3 (since there are 2 odd balls: 1 and 3, out of 3 balls).
• Probability of drawing a K or J card = 2/3 (since there are 2 favorable cards: K and J, out of 3 cards).

Thus, the theoretical probability of both events occurring in a single trial is:

$P(\text{odd ball and K or J card}) = \left(\frac{2}{3}\right) \times \left(\frac{2}{3}\right) = \frac{4}{9}$

You can round this answer to the nearest thousandth, giving you:

$P \approx 0.444$

Part (b) - Experimental Probability:

For the experimental probability, we use the data provided from the trials. You need to find the probability of selecting an odd ball and a K or J card from the given number of trials.

Here’s the summary of the trials:

• Outcome 1K: 62 trials
• Outcome 2K: 70 trials
• Outcome 3K: 61 trials
• Outcome 1J: 68 trials
• Outcome 2J: 64 trials
• Outcome 3J: 56 trials

To calculate the experimental probability, we need to find the total number of trials where an odd ball (1 or 3) was chosen, and a K or J card was drawn.

• Odd balls are 1 and 3, so sum up trials for outcomes 1K, 3K, 1J, and 3J:
  • 1K = 62
  • 3K = 61
  • 1J = 68
  • 3J = 56

Total trials with odd balls and K or J cards = 62 + 61 + 68 + 56 = 247

• Total trials: 550

Thus, the experimental probability is:

$P(\text{odd ball and K or J card}) = \frac{247}{550} \approx 0.449$

Part (c) - Understanding the Difference:

The correct statement regarding the difference between experimental and theoretical probabilities 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 is because experimental results can vary slightly due to random chance, but as the number of trials increases, the experimental probability should approach the theoretical probability.

Would you like further details or have any questions about these calculations? Here are some related questions that can expand your understanding:

1. How does the Law of Large Numbers relate to experimental probability?
2. How would the probabilities change if there were more balls or cards in the experiment?
3. Can experimental probability ever exactly match theoretical probability? Why or why not?
4. How would you interpret a large discrepancy between experimental and theoretical probabilities?
5. How does the number of trials impact the accuracy of the experimental probability?

Tip: When working with probabilities, remember that the theoretical probability assumes perfect randomness, while experimental probability reflects actual outcomes based on real trials.