To find the probability that the first marble chosen is shaded and the second marble chosen is labeled with an odd number, we need to consider each of these events and their probabilities separately, then combine them.
1. Probability that the first marble is shaded:
- Let’s assume there is exactly 1 shaded marble out of the total 11 marbles.
- The probability that the first marble chosen is shaded is calculated as:
[tex]\[
\frac{\text{Number of shaded marbles}}{\text{Total marbles}} = \frac{1}{11}
\][/tex]
2. Probability that the second marble is labeled with an odd number:
- Since we are replacing the marbles after each selection, the total number of marbles remains 11.
- The marbles numbered 1 through 11 give us the following odd numbers: 1, 3, 5, 7, 9, and 11.
- There are 6 odd numbers out of the 11 total marbles.
- The probability that a randomly chosen marble is labeled with an odd number is:
[tex]\[
\frac{\text{Number of odd numbered marbles}}{\text{Total marbles}} = \frac{6}{11}
\][/tex]
3. Combining the probabilities of the two independent events:
- Since the two events are independent (the first marble being shaded does not affect the second marble being odd, because we replace the marbles), we multiply the probabilities:
[tex]\[
\text{Combined probability} = \left(\frac{1}{11}\right) \times \left(\frac{6}{11}\right) = \frac{1 \times 6}{11 \times 11} = \frac{6}{121}
\][/tex]
Thus, the combined probability that the first marble chosen is shaded and the second marble chosen is labeled with an odd number is:
[tex]\[
\frac{6}{121}
\][/tex]
The provided answer choices are:
- [tex]\(\frac{10}{121}\)[/tex]
- [tex]\(\frac{24}{121}\)[/tex]
- [tex]\(\frac{6}{11}\)[/tex]
- [tex]\(\frac{10}{11}\)[/tex]
None of these options match our obtained result of [tex]\(\frac{6}{121}\)[/tex], so there might be a discrepancy or some unstated assumptions in the problem. However, based on our detailed step-by-step solution, [tex]\(\frac{6}{121}\)[/tex] is the correct probability for the given scenario.
