Let's solve this problem step by step to find the probability that the first marble chosen is shaded and the second marble chosen is labeled with an odd number.
1. Total Marbles:
The bag contains 11 equally sized marbles.
2. Probability of Choosing a Shaded Marble First:
Let's assume that there is exactly 1 shaded marble in the bag.
- The probability of choosing the shaded marble in the first draw is [tex]\(\frac{1}{11}\)[/tex].
3. Marbles Labeled with Odd Numbers:
Marbles in the bag are numbered, and numbers range from 1 to 11.
- The odd numbers between 1 and 11 are 1, 3, 5, 7, 9, and 11. There are 6 odd-numbered marbles in total.
4. Probability of Choosing an Odd-Numbered Marble Second:
Since you replace the first marble after drawing it, the total number of marbles remains 11.
- The probability of choosing an odd-numbered marble in the second draw is [tex]\(\frac{6}{11}\)[/tex].
5. Combined Probability:
Since we are dealing with independent events (replacing the marble after each selection), the combined probability of both events happening in sequence is the product of the individual probabilities.
- The combined probability = [tex]\(\left( \frac{1}{11} \right) \times \left( \frac{6}{11} \right)\)[/tex].
6. Calculation:
- The probability of the first marble being shaded is [tex]\(\frac{1}{11}\)[/tex].
- The probability of the second marble being odd-numbered is [tex]\(\frac{6}{11}\)[/tex].
- Combined probability = [tex]\(\frac{1}{11} \times \frac{6}{11} = \frac{6}{121}\)[/tex].
Therefore, the probability that the first marble chosen is shaded and the second marble chosen is labeled with an odd number is:
[tex]\[ \boxed{\frac{6}{121}} \][/tex]
