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A six-sided number cube numbered 1 through 6 is rolled 270 times. The number 3 comes up 44 times.

1. What is the theoretical probability of rolling a 3?
2. What is the experimental probability of rolling a 3?

A. The theoretical probability of rolling a 3 is [tex]\(\frac{1}{6}\)[/tex].
B. The experimental probability of rolling a 3 is [tex]\(\frac{44}{270}\)[/tex].

Sagot :

GinnyAnswer
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].
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