Mali performed an experiment with a standard deck of 52 cards. She wanted to see if the face cards (Ace, King, Queen, and Jack) came up the expected number of times when she randomly selected 13 cards from the deck.

\begin{tabular}{|c|c|c|}
\hline
Card & \begin{tabular}{c}
Theoretical \\ (Predicted \\ Frequency)
\end{tabular} & \begin{tabular}{c}
Experimental \\ (Observed \\ Frequency)
\end{tabular} \\
\hline
Ace & 1 & 0 \\
\hline
King & 1 & 1 \\
\hline
Queen & 1 & 1 \\
\hline
Jack & 1 & 0 \\
\hline
\end{tabular}

Mali noticed that the observed frequencies did not match the predicted frequencies. What does she need to change in her experiment to get closer to the predicted values based on the theoretical probability for all the face cards?



Answer :

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.