To get results that more accurately reflect the theoretical probabilities, Mali needs to increase the sample size in her experiment. Here’s a detailed explanation:
1. Understanding Theoretical Probability for Face Cards:
- In a standard deck of 52 cards, there are 4 suits: hearts, diamonds, clubs, and spades.
- Each suit contains one of each face card (Ace, King, Queen, Jack).
- This means there are 4 Aces, 4 Kings, 4 Queens, and 4 Jacks in the deck.
2. Calculating Theoretical Probability:
- The probability of drawing a specific face card (e.g., Ace) from the full deck:
[tex]\[
P(\text{Ace}) = \frac{\text{Number of Aces}}{\text{Total Number of Cards}} = \frac{4}{52}
\][/tex]
3. Theoretical (Predicted) Frequency:
- If Mali draws a subset of cards, we can calculate the expected (theoretical) frequency by multiplying the total sample size by the theoretical probability.
- For example, with a sample size of 13 cards, the expected number of any specific face card like Ace would be:
[tex]\[
\text{Theoretical Frequency} = 13 \times \frac{4}{52} = 1
\][/tex]
- Originally, this yields an expected frequency of 1 for each face card, which is reflected in the provided theoretical frequencies.
4. Observed Frequencies:
- From her experiment, Mali observed that some face cards appeared more or less frequently than expected:
[tex]\[
\text{Ace: 0, King: 1, Queen: 1, Jack: 0}
\][/tex]
5. Improving Accuracy with a Larger Sample Size:
- Small sample sizes (like 13 cards) can lead to significant deviations from theoretical expectations because each draw’s probability slightly influences the next.
- To get a better approximation of theoretical probabilities, Mali should increase the number of draws. A larger sample size tends to smooth out the random fluctuations and align more closely with theoretical expectations.
- For instance, if she uses a sample size of 52 cards, we can calculate expected frequencies again:
[tex]\[
\text{Theoretical Frequency} = 52 \times \frac{4}{52} = 4
\][/tex]
- This means we would expect each face card (Ace, King, Queen, Jack) to appear approximately 4 times in a larger sample size of 52 cards.
6. Conclusion:
- By increasing her sample size, Mali can expect her experimental results to more closely match the predicted frequencies. This is because larger sample sizes reduce the impact of variance and provide a more accurate reflection of theoretical probabilities.
