A standard deck of 52 cards contains 4 jacks. Jim picks a card from the deck, records the result, puts the card back, and then shuffles the deck. He performs this experiment 1,000 times. He notes that a jack was drawn 80 times. Is the empirical or theoretical probability of drawing a jack higher in this experiment?

To solve this problem, we need to compare the theoretical probability of drawing a jack with the empirical probability observed from Jim's experiment.

### Theoretical Probability
A standard deck of 52 cards contains 4 jacks. The theoretical probability \( P(\text{Jack}) \) of drawing a jack in one trial is calculated as:

\[
P(\text{Jack}) = \frac{\text{Number of Jacks}}{\text{Total Number of Cards}} = \frac{4}{52} = \frac{1}{13} \approx 0.0769
\]

### Empirical Probability
Jim performed the experiment 1,000 times and drew a jack 80 times. The empirical probability \( P_{\text{empirical}}(\text{Jack}) \) is calculated as:

\[
P_{\text{empirical}}(\text{Jack}) = \frac{\text{Number of Times a Jack was Drawn}}{\text{Total Number of Trials}} = \frac{80}{1000} = 0.08
\]

### Comparison
- **Theoretical Probability:** \( \approx 0.0769 \)
- **Empirical Probability:** \( 0.08 \)

The empirical probability (0.08) is slightly higher than the theoretical probability (approximately 0.0769).

### Conclusion
In this experiment, the **empirical probability** of drawing a jack is higher than the theoretical probability.

Would you like more details on this calculation, or do you have any questions?

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Here are 8 related questions you might find interesting:
1. What is the probability of not drawing a jack in one trial?
2. How would the empirical probability change if Jim repeated the experiment 10,000 times?
3. What is the theoretical probability of drawing two jacks in a row (with replacement)?
4. How would the probabilities change if Jim didn't replace the cards after each draw?
5. What is the probability of drawing at least one jack in three trials?
6. How does the law of large numbers relate to the difference between empirical and theoretical probabilities?
7. What is the probability of drawing a king instead of a jack in this scenario?
8. How would the theoretical probability change if the deck had 54 cards instead (including 2 jokers)?

**Tip:** The more trials you perform in an experiment, the closer the empirical probability tends to get to the theoretical probability, due to the law of large numbers.

Explore the comparison between theoretical and empirical probability in drawing jacks from a standard deck of 52 cards. Understand how to calculate probabilities, including the theoretical probability formula and its application in empirical experiments.

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