A set of face cards contains 4 Jacks, 4 Queens, and 4 Kings. Carlie chooses a card from the set, records the type of card, and then replaces the card. She repeats this procedure a total of 60 times. Her results are shown in the table:

\begin{tabular}{|c|c|c|}
\hline \multicolumn{2}{|c|}{Cards} \\
\hline Type of Card & \begin{tabular}{c}
Observed \\
Frequency
\end{tabular} & \begin{tabular}{c}
Relative \\
Frequency
\end{tabular} \\
\hline Jack & 24 & [tex]$\frac{2}{5}$[/tex] \\
\hline Queen & 16 & [tex]$\frac{4}{15}$[/tex] \\
\hline King & 20 & [tex]$\frac{1}{3}$[/tex] \\
\hline
\end{tabular}

How does the experimental probability of choosing a Queen compare with the theoretical probability of choosing a Queen?

The experimental probability is 4 less than the theoretical probability.



Answer :

Let's break down the problem step by step to find how the experimental probability of choosing a Queen compares to the theoretical probability of choosing a Queen.

First, we need to understand the theoretical probability. Since the set of face cards contains 4 Jacks, 4 Queens, and 4 Kings, the total number of cards is:

[tex]\[ 4 + 4 + 4 = 12 \][/tex]

The probability of choosing a Queen from this set is the number of Queens divided by the total number of cards:

[tex]\[ \text{Theoretical Probability of Queen} = \frac{\text{Number of Queens}}{\text{Total Number of Cards}} = \frac{4}{12} = \frac{1}{3} \approx 0.3333 \][/tex]

Next, let's look at the experimental probability. Carlie chooses a card 60 times, and the table shows that she observed 16 Queens. The experimental probability is the number of times a Queen is chosen divided by the total number of draws:

[tex]\[ \text{Experimental Probability of Queen} = \frac{\text{Observed Queens}}{\text{Total Draws}} = \frac{16}{60} = \frac{4}{15} \approx 0.2667 \][/tex]

Now, we need to compare the experimental probability with the theoretical probability. To do this, we subtract the experimental probability from the theoretical probability:

[tex]\[ \text{Difference} = \text{Theoretical Probability} - \text{Experimental Probability} \][/tex]
[tex]\[ \text{Difference} = 0.3333 - 0.2667 = 0.0666 \][/tex]

Thus, the experimental probability of choosing a Queen (0.2667) is less than the theoretical probability of choosing a Queen (0.3333) by approximately 0.0666.

To recap:
1. The theoretical probability of choosing a Queen is approximately 0.3333.
2. The experimental probability of choosing a Queen is approximately 0.2667.
3. The difference between the theoretical and experimental probabilities is approximately 0.0666.