To solve the problem, you need to determine the combined probability of two specific events:
1. Choosing a shaded marble on the first draw.
2. Choosing an odd-numbered marble on the second draw.
Let's break it down step-by-step:
1. First Draw - Probability of Choosing a Shaded Marble:
- There are 11 marbles in total.
- Out of these, 5 are shaded.
- The probability of drawing a shaded marble on the first draw can be calculated by dividing the number of shaded marbles by the total number of marbles:
[tex]\[
\text{Probability of shaded first} = \frac{5}{11}
\][/tex]
2. Second Draw - Probability of Choosing an Odd-Numbered Marble:
- Since the marble is replaced after the first selection, the total number of marbles remains the same.
- There are 11 marbles in total.
- Out of these, 5 are labeled with odd numbers.
- The probability of drawing an odd-numbered marble on the second draw can be calculated by dividing the number of odd-numbered marbles by the total number of marbles:
[tex]\[
\text{Probability of odd numbered second} = \frac{5}{11}
\][/tex]
3. Combined Probability:
- To find the combined probability of both events happening (first drawing a shaded marble and then drawing an odd-numbered marble), you multiply the probabilities of the two independent events:
[tex]\[
\text{Combined probability} = \text{Probability of shaded first} \times \text{Probability of odd numbered second} = \frac{5}{11} \times \frac{5}{11} = \frac{25}{121}
\][/tex]
Therefore, the combined probability that the first marble chosen is shaded and the second marble is labeled with an odd number is [tex]\(\frac{25}{121}\)[/tex].
Now let's compare this to the given options:
- [tex]\(\frac{10}{121}\)[/tex]
- [tex]\(\frac{24}{121}\)[/tex]
- [tex]\(\frac{6}{11}\)[/tex]
- [tex]\(\frac{10}{11}\)[/tex]
Since [tex]\(\frac{25}{121}\)[/tex] is not in the given options, there might have been a mistake in the problem formulation or given options. However, based on our calculations, the correct probability is [tex]\(\frac{25}{121}\)[/tex].
