A clothing store kept track of types and sizes of clothing sold over the period of one week. The two-way table is given below:

\begin{tabular}{|l|c|c|c|c|}
\hline & Small & Medium & Large & Total \\
\hline T-Shirt & 11 & 15 & 8 & 34 \\
\hline Sweatshirt & 6 & 11 & 18 & 35 \\
\hline Sweatpants & 10 & 14 & 7 & 31 \\
\hline Total & 27 & 40 & 33 & 100 \\
\hline
\end{tabular}

What is the probability that a randomly selected clothing item from this table is sweatpants, given the size is small?

[tex]\(P(\text{Sweatpants} \mid \text{Small}) =\)[/tex] [tex]\(\square \square \% \)[/tex]

Round your answer to the nearest whole percent.



Answer :

To solve this problem, we need to determine the probability that a randomly selected clothing item is a pair of sweatpants given that it's of small size. Let's break down the steps involved in finding this probability:

1. Determine the Total Number of Small Size Clothing Items Sold:
According to the table, the total number of small size clothing items sold is 27.

2. Find the Number of Small Size Sweatpants Sold:
From the table, we see that 10 small size sweatpants were sold.

3. Calculate the Conditional Probability:
The conditional probability [tex]\( P(\text{Sweatpants} \mid \text{Small}) \)[/tex] is given by the ratio of the number of small sweatpants to the total number of small size clothing items. This is calculated as:
[tex]\[ P(\text{Sweatpants} \mid \text{Small}) = \frac{\text{Number of Small Sweatpants}}{\text{Total Number of Small Size Clothing Items}} = \frac{10}{27} \][/tex]

4. Convert the Probability to a Percentage:
To express this probability as a percentage, multiply the ratio by 100:
[tex]\[ \text{Probability Percentage} = \left( \frac{10}{27} \right) \times 100 \approx 37.03703703703704 \][/tex]

5. Round to the Nearest Whole Percent:
Finally, we round the percentage to the nearest whole number:
[tex]\[ 37.03703703703704 \approx 37 \][/tex]

Thus, the probability that a randomly selected clothing item is a pair of sweatpants given that it's of small size is approximately 37%.

Therefore,
[tex]\[ P(\text{Sweatpants} \mid \text{Small}) \approx 37 \% \][/tex]