There are 10 brown, 10 black, 10 green, and 10 gold marbles in a bag. A student pulled a marble, recorded the color, and placed the marble back in the bag. The table below lists the frequency of each color pulled during the experiment after 40 trials.

\begin{tabular}{|c|c|}
\hline Outcome & Frequency \\
\hline Brown & 13 \\
\hline Black & 9 \\
\hline Green & 7 \\
\hline Gold & 11 \\
\hline
\end{tabular}

Compare the theoretical probability and experimental probability of pulling a green marble from the bag.

A. The theoretical probability, P(green), is [tex]$50\%$[/tex] and the experimental probability is [tex]$115\%$[/tex].
B. The theoretical probability, P(green), is [tex]$25\%$[/tex], and the experimental probability is [tex]$25\%$[/tex].
C. The theoretical probability, P(green), is [tex]$25\%$[/tex], and the experimental probability is [tex]$17.5\%$[/tex].
D. The theoretical probability, P(green), is [tex]$50\%$[/tex], and the experimental probability is [tex]$7.0\%$[/tex].



Answer :

To tackle this problem, we'll separate it into our two main tasks: calculating the theoretical probability and the experimental probability of pulling a green marble from the bag.

### Theoretical Probability

The theoretical probability is based on the assumption that each marble has an equal chance of being selected.

1. Count the total number of marbles:
- Brown: 10
- Black: 10
- Green: 10
- Gold: 10

Total number of marbles = 10 + 10 + 10 + 10 = 40

2. Find the probability of pulling a green marble:
- There are 10 green marbles out of a total of 40 marbles.

Theoretical Probability [tex]\( P(\text{green}) = \frac{\text{Number of green marbles}}{\text{Total number of marbles}} = \frac{10}{40} = \frac{1}{4} \)[/tex]

3. Convert this into a percentage:
- [tex]\( P(\text{green}) \)[/tex] = [tex]\( \frac{1}{4} \times 100\% = 25\% \)[/tex]

### Experimental Probability

The experimental probability is determined by the results of the actual experiment.

1. Total number of trials:
- According to the table, the number of times each marble was pulled was recorded over 40 trials.

2. Find the frequency of pulling a green marble:
- The table shows that green marbles were pulled 7 times.

3. Calculate the experimental probability:
- Experimental Probability [tex]\( P(\text{green}) = \frac{\text{Frequency of green marbles}}{\text{Total number of trials}} = \frac{7}{40} \)[/tex]

4. Convert this into a percentage:
- [tex]\( P(\text{green}) = \frac{7}{40} \times 100\% = 17.5\% \)[/tex]

### Conclusion

- The theoretical probability of pulling a green marble is 25%.
- The experimental probability of pulling a green marble, based on 40 trials, is 17.5%.

Given this information, you can compare the solutions provided:
1. The theoretical probability, P(green), is [tex]\( 50\% \)[/tex] and the experimental probability is [tex]\( 115\% \)[/tex].
2. The theoretical probability, P(green), is [tex]\( 25\% \)[/tex] and the experimental probability is [tex]\( 25\% \)[/tex].
3. The theoretical probability, P(green), is [tex]\( 25\% \)[/tex] and the experimental probability is [tex]\( 175\% \)[/tex].
4. The theoretical probability, P(green), is [tex]\( 50\% \)[/tex] and the experimental probability is [tex]\( 7.0\% \)[/tex].

The correct statement is:

"The theoretical probability, P(green), is [tex]\( 25\% \)[/tex] and the experimental probability is [tex]\( 17.5\% \)[/tex]."