To determine the probability that a marble chosen at random is either shaded or labeled with a multiple of 3, we can follow a step-by-step approach:
1. Identify the total number of marbles:
There are 11 marbles in the bag.
2. Identify the marbles labeled with multiples of 3:
The multiples of 3 within the range of 1 to 11 are:
- 3
- 6
- 9
Therefore, there are 3 marbles that are labeled with a multiple of 3.
3. Identify the shaded marbles:
Let's assume that the indices of the shaded marbles are 2 and 4.
4. Combine the sets of shaded marbles and marbles labeled with multiples of 3:
We need to consider the union of these two sets, ensuring there are no duplications.
5. List of all unique desired marbles:
- Marbles labeled with multiples of 3: {3, 6, 9}
- Shaded marbles: {2, 4}
Combining these without duplication gives us the set:
{2, 3, 4, 6, 9}
Therefore, the total number of unique marbles that are either shaded or labeled with a multiple of 3 is 5.
6. Determine the probability:
The probability of selecting one of these desired marbles out of the total 11 marbles is:
[tex]\[
\frac{\text{Number of desired marbles}}{\text{Total number of marbles}} = \frac{5}{11}
\][/tex]
Hence, the probability that a randomly chosen marble is either shaded or labeled with a multiple of 3 is [tex]\(\boxed{\frac{5}{11}}\)[/tex].
