To solve for the probability that a randomly chosen customer has purchased either a watch or a repair, we will use the principle of inclusion and exclusion in probability.
Let's start by analyzing the data provided in the table:
[tex]\[
\begin{tabular}{|l|c|c|c|}
\hline & Remodel & Repair & \begin{tabular}{c} New \\ Purchase \end{tabular} \\
\hline Watch & 73 & 47 & 19 \\
\hline Clock & 61 & 59 & 11 \\
\hline Alarm Clock & 83 & 41 & 17 \\
\hline
\end{tabular}
\][/tex]
### Step 1: Calculate the total number of purchases
We sum all the values in the table:
For Watches:
[tex]\[ 73 (\text{Remodel}) + 47 (\text{Repair}) + 19 (\text{New Purchase}) = 139 \][/tex]
For Clocks:
[tex]\[ 61 (\text{Remodel}) + 59 (\text{Repair}) + 11 (\text{New Purchase}) = 131 \][/tex]
For Alarm Clocks:
[tex]\[ 83 (\text{Remodel}) + 41 (\text{Repair}) + 17 (\text{New Purchase}) = 141 \][/tex]
Total purchases:
[tex]\[ 139 + 131 + 141 = 411 \][/tex]
### Step 2: Calculate the total number of watch purchases
We sum up all the purchase types for watches:
[tex]\[ 73 (\text{Remodel}) + 47 (\text{Repair}) + 19 (\text{New Purchase}) = 139 \][/tex]
### Step 3: Calculate the total number of repair purchases
We sum up all the repairs across all categories:
[tex]\[ 47 (\text{Watch}) + 59 (\text{Clock}) + 41 (\text{Alarm Clock}) = 147 \][/tex]
### Step 4: Apply the principle of inclusion and exclusion
We need to find the total number of cases which are either a watch purchase or a repair. If we add the total watch purchases and the total repair purchases, we end up double-counting the cases where watches were repaired. So, we need to subtract these overlapping cases:
Total number of watch purchases:
[tex]\[ 139 \][/tex]
Total number of repair purchases:
[tex]\[ 147 \][/tex]
Overlap (watch repairs):
[tex]\[ 47 \][/tex]
Number of cases which are either a watch purchase or a repair:
[tex]\[ 139 + 147 - 47 = 239 \][/tex]
### Step 5: Calculate the probability
The probability is the number of favorable outcomes divided by the total number of possible outcomes (total purchases):
[tex]\[ P(\text{Watch or Repair}) = \frac{\text{Number of Watch or Repair cases}}{\text{Total Purchases}} = \frac{239}{411} \][/tex]
So, the probability that a randomly chosen customer has purchased a watch or a repair is:
[tex]\[ P(\text{Watch or Repair}) = \frac{239}{411} \approx 0.5815 \][/tex]
In simplest fractional form, the answer remains [tex]\( \frac{239}{411} \)[/tex] as it cannot be simplified further. Thus, the decimal approximation is approximately:
[tex]\[
\boxed{0.5815}
\][/tex]
