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To address the problem of determining both the theoretical and experimental probabilities of rolling a 3 on a six-sided number cube, let's break down the calculations step-by-step:
### Theoretical Probability
1. Definition: The theoretical probability of an event is calculated based on the known possible outcomes, assuming each outcome is equally likely.
2. Possible Outcomes on a Six-Sided Die: When rolling a fair six-sided die, there are 6 possible outcomes (1, 2, 3, 4, 5, and 6).
3. Desired Outcome: We are interested in the outcome where the die shows a 3.
Using these points,
[tex]\[ \text{Theoretical Probability} = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}} \][/tex]
Thus,
[tex]\[ \text{Theoretical Probability of rolling a 3} = \frac{1}{6} \][/tex]
### Experimental Probability
1. Definition: The experimental probability is calculated based on the actual outcomes observed from conducting an experiment.
2. Total Rolls: According to the problem, the die was rolled 270 times.
3. Number of Times a 3 Appeared: The number 3 came up 44 times in these 270 rolls.
Using these points,
[tex]\[ \text{Experimental Probability} = \frac{\text{Number of Times the Event Occurred}}{\text{Total Number of Trials}} \][/tex]
Hence,
[tex]\[ \text{Experimental Probability of rolling a 3} = \frac{44}{270} \][/tex]
Now let's summarize:
- The theoretical probability of rolling a 3 is [tex]\( \frac{1}{6} \)[/tex], which is approximately 0.1667 when converted to decimal form.
- The experimental probability of rolling a 3 based on the experimental data is [tex]\( \frac{44}{270} \)[/tex], which simplifies to around 0.16296 when converted to decimal form.
Comparing these to the provided results:
- The theoretical probability [tex]\( \frac{1}{6} \)[/tex] corresponds to approximately 0.1667.
- The experimental probability [tex]\( \frac{44}{270} \)[/tex] corresponds to approximately 0.16296.
### Correct Answer Choices:
Based on these calculations, the correct answers are:
- The theoretical probability of rolling a 3 is [tex]\( \frac{1}{6} \)[/tex].
- The experimental probability of rolling a 3 is [tex]\( \frac{44}{270} \)[/tex].
Since [tex]\( \frac{44}{270} \)[/tex] simplifies to [tex]\( \frac{22}{135} \)[/tex], another correct choice for the experimental probability would be:
- The experimental probability of rolling a 3 is [tex]\( \frac{22}{135} \)[/tex].
### Theoretical Probability
1. Definition: The theoretical probability of an event is calculated based on the known possible outcomes, assuming each outcome is equally likely.
2. Possible Outcomes on a Six-Sided Die: When rolling a fair six-sided die, there are 6 possible outcomes (1, 2, 3, 4, 5, and 6).
3. Desired Outcome: We are interested in the outcome where the die shows a 3.
Using these points,
[tex]\[ \text{Theoretical Probability} = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}} \][/tex]
Thus,
[tex]\[ \text{Theoretical Probability of rolling a 3} = \frac{1}{6} \][/tex]
### Experimental Probability
1. Definition: The experimental probability is calculated based on the actual outcomes observed from conducting an experiment.
2. Total Rolls: According to the problem, the die was rolled 270 times.
3. Number of Times a 3 Appeared: The number 3 came up 44 times in these 270 rolls.
Using these points,
[tex]\[ \text{Experimental Probability} = \frac{\text{Number of Times the Event Occurred}}{\text{Total Number of Trials}} \][/tex]
Hence,
[tex]\[ \text{Experimental Probability of rolling a 3} = \frac{44}{270} \][/tex]
Now let's summarize:
- The theoretical probability of rolling a 3 is [tex]\( \frac{1}{6} \)[/tex], which is approximately 0.1667 when converted to decimal form.
- The experimental probability of rolling a 3 based on the experimental data is [tex]\( \frac{44}{270} \)[/tex], which simplifies to around 0.16296 when converted to decimal form.
Comparing these to the provided results:
- The theoretical probability [tex]\( \frac{1}{6} \)[/tex] corresponds to approximately 0.1667.
- The experimental probability [tex]\( \frac{44}{270} \)[/tex] corresponds to approximately 0.16296.
### Correct Answer Choices:
Based on these calculations, the correct answers are:
- The theoretical probability of rolling a 3 is [tex]\( \frac{1}{6} \)[/tex].
- The experimental probability of rolling a 3 is [tex]\( \frac{44}{270} \)[/tex].
Since [tex]\( \frac{44}{270} \)[/tex] simplifies to [tex]\( \frac{22}{135} \)[/tex], another correct choice for the experimental probability would be:
- The experimental probability of rolling a 3 is [tex]\( \frac{22}{135} \)[/tex].
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