Question

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:

type of card	jack	queen	king
relative frequency	2/5	4/15	1/3

how does the experimental probability of choosing a queen compare with the theoretical probability of choosing a queen?
options:

• the experimental probability is 4 less than the theoretical probability.
• the experimental probability is 1/15 less than the theoretical probability.

Explanation:

Step1: Find total face cards

There are 4 Jacks, 4 Queens, and 4 Kings, so total face cards \( n = 4 + 4 + 4 = 12 \).

Step2: Calculate theoretical probability of Queen

Theoretical probability \( P_{theoretical} = \frac{\text{Number of Queens}}{\text{Total face cards}} = \frac{4}{12} = \frac{1}{3} \).

Step3: Calculate experimental probability of Queen

From the table, observed frequency of Queen is 16, total trials \( N = 60 \). Experimental probability \( P_{experimental} = \frac{16}{60} = \frac{4}{15} \).

Step4: Find the difference

Difference \( = P_{theoretical} - P_{experimental} = \frac{1}{3} - \frac{4}{15} \). Convert \( \frac{1}{3} \) to \( \frac{5}{15} \), so \( \frac{5}{15} - \frac{4}{15} = \frac{1}{15} \). So experimental probability is \( \frac{1}{15} \) less than theoretical.

Answer:

The experimental probability is \(\frac{1}{15}\) less than the theoretical probability. (Corresponding to the option "The experimental probability is \(\frac{1}{15}\) less than the theoretical probability.")