Answer the question based on the data in the table.

\begin{tabular}{|c|c|c|c|}
\hline
\multirow{2}{*}{\begin{tabular}{c}
Shirt \\
Color
\end{tabular}} & \multicolumn{3}{|c|}{ Size } \\
\cline { 2 - 4 } & Large & Medium & Total \\
\hline Red & 42 & 48 & 90 \\
\hline Blue & 35 & 40 & 75 \\
\hline Total & 77 & 88 & 165 \\
\hline
\end{tabular}

Select the correct answer.

If you pick a shirt at random from the given batch of 165 shirts, what is the probability that it is red and the size is medium?

A. [tex]$\frac{90}{165}$[/tex]
B. [tex]$\frac{88}{165}$[/tex]
C. [tex]$\frac{48}{27225}$[/tex]
D. [tex]$\frac{90}{27225}$[/tex]
E. [tex]$\frac{48}{165}$[/tex]



Answer :

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]