To find the probability that you and your friend both get lemon-lime bottles, we need to consider the sequence of events step-by-step.
1. Calculate the total number of bottles initially:
- There are 4 lemon-lime bottles.
- There are 7 orange bottles.
- Total bottles = [tex]\(4 + 7 = 11\)[/tex].
2. Calculate the probability of drawing a lemon-lime bottle first:
- The probability of drawing a lemon-lime bottle first is:
[tex]\[
\frac{\text{Number of lemon-lime bottles initially}}{\text{Total number of bottles initially}} = \frac{4}{11}.
\][/tex]
3. Update the number of bottles after the first draw:
- After taking one lemon-lime bottle, the remaining bottles are:
- Lemon-lime bottles left = [tex]\(4 - 1 = 3\)[/tex].
- Total bottles left = [tex]\(11 - 1 = 10\)[/tex].
4. Calculate the probability of drawing a lemon-lime bottle second:
- The probability of drawing a lemon-lime bottle second (after one has already been taken) is:
[tex]\[
\frac{\text{Remaining number of lemon-lime bottles}}{\text{Remaining total number of bottles}} = \frac{3}{10}.
\][/tex]
5. Calculate the combined probability of both events occurring:
- The probability that both events occur (both you and your friend getting a lemon-lime bottle) is the product of the two probabilities:
[tex]\[
\left(\frac{4}{11}\right) \times \left(\frac{3}{10}\right) = \frac{4 \cdot 3}{11 \cdot 10} = \frac{12}{110} = \frac{6}{55}.
\][/tex]
Hence, the probability that both you and your friend get lemon-lime bottles is [tex]\(\frac{6}{55}\)[/tex], which corresponds to answer choice A.
