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.
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:
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.
