To solve this problem, we need to find the probability that a randomly selected shirt from the batch of 165 shirts is red and medium-sized.
Given:
- The total number of shirts is 165.
- The number of red medium shirts is 48.
The probability of an event is given by the formula:
[tex]\[ P(\text{Event}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \][/tex]
In this case, the event we are interested in is selecting a red medium-sized shirt.
The number of favorable outcomes (red medium shirts) is 48, and the total number of possible outcomes (total shirts) is 165.
So, the probability [tex]\( P \)[/tex] is:
[tex]\[ P(\text{red and medium}) = \frac{48}{165} \][/tex]
We simplify [tex]\( \frac{48}{165} \)[/tex] to a decimal for clarity:
[tex]\[ P(\text{red and medium}) = 0.2909090909090909 \][/tex]
Since [tex]\( \frac{48}{165} \)[/tex] corresponds to option C when simplified:
Therefore, the correct answer is:
[tex]\[ \boxed{\frac{48}{165}} \][/tex]
