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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:

| | 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 |

What is the probability that a randomly selected clothing item from this table is a t-shirt, given the size is large?

[tex]\[ P(\text{T-Shirt | Large}) = [?]\% \][/tex]

Round your answer to the nearest whole percent.

Sagot :

To address the question of determining the probability that a randomly selected clothing item is a T-shirt given that it is large, we need to follow these steps:

1. Identify the total number of large clothing items sold.
From the table, we see:
- Total large clothing items = [tex]\(33\)[/tex]

2. Identify the number of large T-shirts sold.
From the table, we see:
- Number of large T-shirts sold = [tex]\(8\)[/tex]

3. Calculate the conditional probability [tex]\( P(\text{T-Shirt | Large}) \)[/tex]:
The probability of an event given conditioning on another event is calculated by dividing the number of favorable outcomes by the total number of outcomes under the given condition.

[tex]\[ P(\text{T-Shirt | Large}) = \frac{\text{Number of large T-shirts}}{\text{Total number of large items}} \][/tex]

Substituting the values we have:
[tex]\[ P(\text{T-Shirt | Large}) = \frac{8}{33} \][/tex]

4. Convert this probability into a percentage:
To express this probability as a percentage, we multiply by 100:
[tex]\[ P(\text{T-Shirt | Large}) = \left( \frac{8}{33} \right) \times 100 \][/tex]
The result of this calculation is approximately [tex]\(24.242424242424242\%\)[/tex].

5. Round the percentage to the nearest whole number:
Hence, [tex]\(24.242424242424242\%\)[/tex] rounds to [tex]\(24\%\)[/tex].

Therefore, the probability that a randomly selected clothing item from this store is a T-shirt, given that it is large, is:
[tex]\[ P(\text{T-Shirt | Large}) \approx 24\% \][/tex]

Thus, the final answer is:
[tex]\[ 24\% \][/tex]