At a gift shop, the purchases for one week are recorded in the table below:

\begin{tabular}{|l|c|c|c|}
\hline & Tee Shirt & \begin{tabular}{c}
Long Sleeve \\
Shirt
\end{tabular} & Jacket \\
\hline Small & 71 & 38 & 2 \\
\hline Medium & 97 & 41 & 23 \\
\hline Large & 31 & 11 & 0 \\
\hline
\end{tabular}

If we choose a customer at random, what is the probability that they have purchased a jacket and it is medium?

[tex]\[ P (\text{Jacket and Medium}) = \][/tex]

Give your answer in simplest form.



Answer :

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].