To determine the probability that a randomly selected customer purchased a medium-sized jacket, we need to first calculate two key quantities: the total number of purchases and the number of purchases of medium-sized jackets.
From the provided table, the number of purchases is distributed in the following manner:
- Small:
- Tee Shirt: 71
- Long Sleeve Shirt: 38
- Jacket: 2
- Medium:
- Tee Shirt: 97
- Long Sleeve Shirt: 41
- Jacket: 23
- Large:
- Tee Shirt: 31
- Long Sleeve Shirt: 11
- Jacket: 0
First, we'll find the total number of purchases by summing up all the values in the table:
[tex]\[
71 + 38 + 2 + 97 + 41 + 23 + 31 + 11 + 0 = 314
\][/tex]
Thus, the total number of purchases is 314.
Next, we identify the number of medium-sized jackets purchased, which is already given in the table as 23.
The probability of randomly selecting a customer who purchased a medium-sized jacket is given by the ratio of the number of medium-sized jackets to the total number of purchases.
[tex]\[
P (\text{Medium and Jacket}) = \frac{\text{Number of Medium Jackets}}{\text{Total Purchases}} = \frac{23}{314}
\][/tex]
This fraction represents the probability in its simplest form given the context of the problem. Simplifying [tex]\(\frac{23}{314}\)[/tex] further does not change that it is the lowest term possible, which is:
[tex]\[
\boxed{\frac{23}{314}}
\][/tex]
So, the probability that a randomly chosen customer purchased a medium-sized jacket is [tex]\(\frac{23}{314}\)[/tex].
