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 1050 trials. Answer each part. (a) 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 K card, in a single trial. Round your answer to the nearest thousandth. (b) Use the data to find the experimental probability of this event: both choosing the 1 or 2 ball and drawing the K card, in a single trial. Round your answer to the nearest thousandth. (c) Choose the statement that is true. With a large number of trials, there must be a large difference between the experimental and theoretical probabilities. With a large number of trials, there might be a difference between the experimental and theoretical probabilities, but the difference should be small. With a large number of trials, there must be no difference between the experimental and theoretical probabilities.

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 1050 trials. Answer each part. (a) 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 K card, in a single trial. Round your answer to the nearest thousandth. (b) Use the data to find the experimental probability of this event: both choosing the 1 or 2 ball and drawing the K card, in a single trial. Round your answer to the nearest thousandth. (c) Choose the statement that is true. With a large number of trials, there must be a large difference between the experimental and theoretical probabilities. With a large number of trials, there might be a difference between the experimental and theoretical probabilities, but the difference should be small. With a large number of trials, there must be no difference between the experimental and theoretical probabilities.
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 1050 trials. Answer each part. (a) 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 $K$ card, in a single trial. Round your answer to the nearest thousandth. $\square$ (b) Use the data to find the experimental probability of this event: both choosing the 1 or 2 ball and drawing the $K$ card, in a single trial. Round your answer to the nearest thousandth. $\square$ (c) Choose the statement that is true. With a large number of trials, there must be a large difference between the experimental and theoretical probabilities. With a large number of trials, there might be a difference between the experimental and theoretical probabilities, but the difference should be small. With a large number of trials, there must be no difference between the experimental and theoretical probabilities.
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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 1050 total trials and 228 trials resulting in the specific outcomes of interest, the experimental probability is $\frac{228}{1050} = 0.217$.

Step 3: Comparison and Conclusion

The experimental probability (0.217) 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: 0.217

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