Questions: There is a bag with three balls numbered 1 to 3. There is also a pack of three cards lettered K, Q, and J. As a trial of an experiment, a ball was chosen and a card drawn. The number (1 to 3) of the ball and the letter (K, Q, or J) of the card drawn were recorded. Here is a summary of the data from 1375 trials. 1 K 2 K 152 156 2 K 3 K 156 153 1 3 K 1 153 147 1 Q 2 Q 147 157 2 Q 3 Q 157 150 15 1 J 2 J 159 155 3 J 146 Answer each part. Assuming the ball was chosen and the card was drawn at random, find the theoretical probability of this event: both choosing the 1 or 2 ball and drawing the Q card, in a single trial. Round your answer to the nearest thousandth. (II) (b) Use the data to find the experimental probability of this event: both choosing the 1 or 2 ball and drawing the Q card, in a single trial. Round your answer to the nearest thousandth. (c) Choose the statement that is true. - The experimental probability will never be very close to the theoretical probability, no matter the number of trials - The larger the number of trials, the greater the likelihood that the experimental probability will be close to the theoretical probability - The smaller the number of trials, the greater the likelihood that the experimental probability will be close to the theoretical probability.

There is a bag with three balls numbered 1 to 3. There is also a pack of three cards lettered K, Q, and J. As a trial of an experiment, a ball was chosen and a card drawn. The number (1 to 3) of the ball and the letter (K, Q, or J) of the card drawn were recorded. Here is a summary of the data from 1375 trials. 1 K 2 K 152 156 2 K 3 K 156 153 1 3 K 1 153 147 1 Q 2 Q 147 157 2 Q 3 Q 157 150 15 1 J 2 J 159 155 3 J 146 Answer each part. Assuming the ball was chosen and the card was drawn at random, find the theoretical probability of this event: both choosing the 1 or 2 ball and drawing the Q card, in a single trial. Round your answer to the nearest thousandth. (II) (b) Use the data to find the experimental probability of this event: both choosing the 1 or 2 ball and drawing the Q card, in a single trial. Round your answer to the nearest thousandth. (c) Choose the statement that is true. - The experimental probability will never be very close to the theoretical probability, no matter the number of trials - The larger the number of trials, the greater the likelihood that the experimental probability will be close to the theoretical probability - The smaller the number of trials, the greater the likelihood that the experimental probability will be close to the theoretical probability.
Transcript text: There is a bag with three balls numbered 1 to 3. There is also a pack of three cards lettered $K, Q$, and $J$. As a trial of an experiment, a ball was chosen and a card drawn. The number $(1$ to 3) of the ball and the letter $(K, Q$, or $J)$ of the card drawn were recorded. Here is a summary of the data from 1375 trials. \begin{tabular}{l|l|l} $1 K$ & $2 K$ \\ \hline 152 & 156 & \\ \end{tabular} \begin{tabular}{|l|l|l} $2 K$ & $3 K$ \\ \hline 156 & 153 & 1 \\ \hline \end{tabular} \begin{tabular}{|l|l|l} \hline $3 K$ & 1 \\ \hline 153 & 147 \\ \hline \end{tabular} \begin{tabular}{|l|l} $1 Q$ & $2 Q$ \\ \hline 147 & 157 \\ \hline \end{tabular} \begin{tabular}{|l|l|l} \hline $2 Q$ & $3 Q$ \\ \hline 157 & 150 & 15 \\ \hline \end{tabular} \begin{tabular}{|l|l|} \hline $1 J$ & $2 J$ \\ \hline 159 & 155 \\ \hline \end{tabular} \begin{tabular}{l|l|} $3 J$ \\ 146 \\ \hline \end{tabular} Answer each part. Assuming the ball was chosen and the card was drawn at random, find the theoretical probability of this event: both choosing the 1 or 2 ball and drawing the $Q$ card, in a single trial. Round your answer to the nearest thousandth. (II) (b) Use the data to find the experimental probability of this event: both choosing the 1 or 2 ball and drawing the $Q$ card, in a single trial. Round your answer to the nearest thousandth. (c) Choose the statement that is true. - The experimental probability will never be very close to the theoretical probability, no matter the number of trials - The larger the number of trials, the greater the likelihood that the experimental probability will be close to the theoretical probability - The smaller the number of trials, the greater the likelihood that the experimental probability will be close to the theoretical probability.
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Solution

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Solution Steps

Step 1: Theoretical Probability Calculation

Given that the first random process has 3 outcomes and the specific outcome of interest is 2, the probability of the specific outcome in the first process is $\frac{2}{3} = 0.667$. Similarly, for the second random process with 3 outcomes and 1 as the specific outcome of interest, the probability is $\frac{1}{3} = 0.333$. Assuming independence, the theoretical probability of both events occurring is $(0.667) \times (0.333) = 0.222$.

Step 2: Experimental Probability Calculation

With 1375 total trials and 304 trials resulting in the specific outcomes of interest, the experimental probability is $\frac{304}{1375} = 0.221$.

Step 3: Comparison and Conclusion

The experimental probability (0.221) is less than the theoretical probability (0.222). This comparison illustrates how the experimental probability approaches the theoretical probability as the number of trials increases, demonstrating the Law of Large Numbers. The more trials conducted, the closer the experimental probability is expected to be to the theoretical probability.

Final Answer:

Theoretical Probability: 0.222, Experimental Probability: 0.221

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