Let's solve this step-by-step.
1. Determine the probability that the first marble chosen is shaded.
- There are a total of 11 marbles.
- Out of these 11 marbles, 5 are shaded.
- Hence, the probability that the first marble chosen is shaded is [tex]\( \frac{5}{11} \)[/tex].
2. Determine the probability that the second marble chosen is labeled with an odd number.
- Since the marbles are replaced after each selection, there are still 11 marbles to choose from for the second selection.
- Out of the 11 marbles, 6 are labeled with an odd number.
- Therefore, the probability that the second marble chosen is labeled with an odd number is [tex]\( \frac{6}{11} \)[/tex].
3. Determine the combined probability.
- The probability of two independent events both occurring is the product of their individual probabilities.
- Thus, the combined probability is:
[tex]\[
\left( \frac{5}{11} \right) \times \left( \frac{6}{11} \right) = \frac{5 \times 6}{11 \times 11}
\][/tex]
Simplifying the numerator and denominator, we get:
[tex]\[
\frac{30}{121}
\][/tex]
4. Conclusion
- So, the probability that the first marble chosen is shaded and the second marble chosen is labeled with an odd number is [tex]\( \frac{30}{121} \)[/tex].
- Comparing this with the given options, we notice that none of the options ( [tex]\(\frac{10}{121}\)[/tex], [tex]\(\frac{24}{121}\)[/tex], [tex]\(\frac{6}{11}\)[/tex], [tex]\(\frac{10}{11}\)[/tex] ) match our calculated probability.
As a result, none of the given options correctly represent the calculated probability of [tex]\(\frac{30}{121}\)[/tex].
