To determine the probability that a randomly selected clothing item is sweatpants, given that the size is small, we use conditional probability. Conditional probability is calculated by dividing the number of favorable outcomes by the total number of outcomes for the given condition.
Here’s the step-by-step solution to the problem:
1. Identify the total number of small-sized clothing items:
According to the table:
- T-Shirts (Small): 11
- Sweatshirts (Small): 6
- Sweatpants (Small): 10
Therefore, the total number of small-sized clothing items is:
[tex]\[
11 + 6 + 10 = 27
\][/tex]
2. Identify the number of small-sized sweatpants:
According to the table, the number of small-sized sweatpants is 10.
3. Calculate the conditional probability:
The conditional probability that the clothing item is sweatpants given that it is small is calculated as:
[tex]\[
P(\text{Sweatpants} \mid \text{Small}) = \frac{\text{Number of small-sized sweatpants}}{\text{Total number of small-sized clothing items}}
\][/tex]
Substituting the known values:
[tex]\[
P(\text{Sweatpants} \mid \text{Small}) = \frac{10}{27}
\][/tex]
4. Convert the fraction to a percentage:
[tex]\[
P(\text{Sweatpants} \mid \text{Small}) = \left( \frac{10}{27} \right) \times 100 \approx 37.04\%
\][/tex]
5. Round to the nearest whole percent:
[tex]\[
P(\text{Sweatpants} \mid \text{Small}) \approx 37\%
\][/tex]
Therefore, the probability that a randomly selected clothing item is sweatpants, given that the size is small, is approximately [tex]\( 37\% \)[/tex].
