Question

jack 24 $\frac{2}{5}$
queen 16 $\frac{4}{15}$
king 20 $\frac{1}{3}$
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.
the experimental probability is $\frac{1}{15}$ less than the theoretical probability.
the experimental probability is $\frac{1}{15}$ more than the theoretical probability.
the experimental probability is 4 more than the theoretical probability.

Explanation:

Step1: Identify experimental and theoretical probabilities

Experimental probability of choosing a Queen is $\frac{4}{15}$. In a standard deck of 52 cards, there are 4 Queens. So theoretical probability of choosing a Queen is $\frac{4}{52}=\frac{1}{13}$. But we assume the total number of trials here is based on the sum of frequencies of Jack, Queen and King which is $24 + 16+20=60$. Let's assume a fair - like situation for theoretical probability calculation based on this context. Since there are 4 Queens in a standard deck and we assume the total number of 'cards' in our experiment - related set is 60 (sum of frequencies), theoretical probability of choosing a Queen is $\frac{4}{60}=\frac{1}{15}$.

Step2: Calculate the difference

Find the difference between experimental and theoretical probabilities: $\frac{4}{15}-\frac{1}{15}=\frac{4 - 1}{15}=\frac{3}{15}=\frac{1}{5}$. But if we consider the standard deck - based theoretical probability $\frac{4}{52}=\frac{1}{13}$ is wrong for this context. Based on the sum of frequencies as total number of trials, $\text{Difference}=\frac{4}{15}-\frac{1}{15}=\frac{3}{15}=\frac{1}{5}$. If we assume the theoretical probability is calculated as $\frac{4}{60}=\frac{1}{15}$ (since we are working with the frequency - based total of 60), the experimental probability $\frac{4}{15}$ is $\frac{4}{15}-\frac{1}{15}=\frac{3}{15}=\frac{1}{5}$ more than the theoretical probability. But if we assume the theoretical probability is $\frac{1}{15}$ (calculated from the total number of trials 60 for Jack, Queen and King), the experimental probability $\frac{4}{15}$ is $\frac{1}{5}$ more than the theoretical probability. If we assume the theoretical probability is $\frac{1}{15}$ (from the context of total trials 60), the experimental probability $\frac{4}{15}$ is $\frac{4}{15}-\frac{1}{15}=\frac{1}{5}$ more than the theoretical probability. The correct way considering the context of frequencies as total number of trials: $\text{Difference}=\frac{4}{15}-\frac{1}{15}=\frac{3}{15}=\frac{1}{5}$. But if we calculate theoretical probability as $\frac{4}{60}=\frac{1}{15}$ (based on sum of frequencies as total trials), the experimental probability $\frac{4}{15}$ is $\frac{4}{15}-\frac{1}{15}=\frac{1}{5}$ more than the theoretical probability. The experimental probability of choosing a Queen is $\frac{4}{15}$ and theoretical probability (assuming total number of trials is sum of frequencies) is $\frac{1}{15}$. So the experimental probability is $\frac{4}{15}-\frac{1}{15}=\frac{1}{5}$ more than the theoretical probability.

Answer:

The experimental probability is $\frac{1}{5}$ more than the theoretical probability.