Certainly! Let’s go through the problem step by step.
### Understanding the Problem:
We are given the following two-way table that shows the number of different types and sizes of clothing items sold over one week:
| | Small | Medium | Large | Total |
|-------------|-------|--------|-------|-------|
| T-Shirt | 11 | 15 | 8 | 34 |
| Sweatshirt | 6 | 11 | 18 | 35 |
| Sweatpants | 10 | 14 | 7 | 31 |
| Total | 27| 40 | 33| 100|
We want to find the probability that a randomly selected clothing item is sweatpants given that the size is small. Symbolically, this is written as:
[tex]\[ P(\text{Sweatpants} \mid \text{Small}) \][/tex]
### Step-by-Step Solution:
1. Identify the required data:
- The total number of small clothing items sold (denoted as [tex]\( \text{Total Small} \)[/tex])
- The number of small sweatpants sold ([tex]\( \text{Small Sweatpants} \)[/tex])
2. Extract Data from the Table:
- Total Small: According to the table, the total number of small items sold is 27.
- Small Sweatpants: The number of small sweatpants sold is 10.
3. Calculate the Probability:
The probability that a randomly selected clothing item is sweatpants given that the size is small can be calculated using the formula for conditional probability:
[tex]\[
P(\text{Sweatpants} \mid \text{Small}) = \frac{\text{Number of Small Sweatpants}}{\text{Total Number of Small Items}}
\][/tex]
Substitute the values obtained:
[tex]\[
P(\text{Sweatpants} \mid \text{Small}) = \frac{10}{27}
\][/tex]
4. Convert to Percentage:
To express the probability as a percentage, multiply the result by 100:
[tex]\[
P(\text{Sweatpants} \mid \text{Small}) = \left(\frac{10}{27}\right) \times 100 \approx 37.037\%
\][/tex]
### Final Answer:
The probability that a randomly selected clothing item is sweatpants given that the size is small is approximately [tex]\( \boxed{37.037\%} \)[/tex].
