To address the problem, we'll need to analyze the differences between the experimental and theoretical probabilities of rolling a 3 using the given frequencies.
Step-by-Step Solution:
1. Determine the Total Number of Rolls:
First, sum up the frequencies of each rolled number to find the total number of rolls.
[tex]\[
16 (for\ 2) + 24 (for\ 3) + 26 (for\ 4) + 16 (for\ 5) + 20 (for\ 6) = 102
\][/tex]
So, the total number of rolls is [tex]\( 102 \)[/tex].
2. Find the Experimental Probability of Rolling a 3:
The frequency of rolling a 3 is [tex]\( 24 \)[/tex]. Therefore, the experimental probability [tex]\( P_{exp}(3) \)[/tex] is:
[tex]\[
P_{exp}(3) = \frac{24}{102}
\][/tex]
3. Find the Theoretical Probability of Rolling a 3:
Assuming we have a fair six-sided die, the theoretical probability [tex]\( P_{theo}(3) \)[/tex] is:
[tex]\[
P_{theo}(3) = \frac{1}{6}
\][/tex]
4. Convert Probabilities to Common Denominator to Compare (Optional but Useful):
Since [tex]\( \frac{1}{6} = \frac{17}{102} \)[/tex] to make a fair comparison (since [tex]\( P_{exp}(3) \)[/tex] was [tex]\( 102 \)[/tex] total rolls), we can convert the theoretical probability to the same denominator:
[tex]\[
P_{theo}(3) = \frac{1}{6} = \frac{17}{102}
\][/tex]
5. Compare Experimental and Theoretical Probabilities:
Now, subtract the theoretical probability from the experimental probability to examine the difference:
[tex]\[
\frac{24}{102} - \frac{17}{102} = \frac{7}{102}
\][/tex]
6. Interpret the Difference:
We analyze whether this result matches any of the given options:
[tex]\[
\frac{7}{102} \quad ?= \quad \frac{1}{30}, \quad -\frac{1}{30}, \quad \frac{2}{3} \times \frac{1}{6}, \quad -\frac{2}{3} \times \frac{1}{6}
\][/tex]
- [tex]\( \frac{1}{30} = \frac{3.4}{102} \)[/tex]
- [tex]\( 2/3 \times 1/6 = \frac{2}{18} = \frac{1}{9} = \frac{11.3}{102} \)[/tex]
None of these match exactly:
- [tex]\( \frac{1}{30} = 0.0333 \)[/tex]
- [tex]\( \frac{2/3 \square \times 1/6}{=0.1111} \)[/tex]
The experimentally found result is [tex]\( \frac{7}{102} = 0.0686 \)[/tex].
Conclusion:
Given the available options, none of the comparisons match the calculated difference [tex]\( \frac{7}{102} \)[/tex] ([tex]\(0.0686\)[/tex]) we found experimentally. Therefore, the experimental and theoretical probabilities comparison does not fit any of the listed specific descriptions.
