To find the probability that a randomly chosen customer has purchased an alarm clock or a new purchase, we need to follow several steps:
1. Determine the Total Number of Purchases:
We sum all the purchases recorded in the given table.
[tex]\[
\begin{aligned}
\text{Total Purchases} &= (\text{Remodel Watch} + \text{Repair Watch} + \text{New Purchase Watch}) \\
&\quad + (\text{Remodel Clock} + \text{Repair Clock} + \text{New Purchase Clock}) \\
&\quad + (\text{Remodel Alarm Clock} + \text{Repair Alarm Clock} + \text{New Purchase Alarm Clock}) \\
&= 73 + 47 + 19 + 61 + 59 + 11 + 83 + 41 + 17 \\
&= 411
\end{aligned}
\][/tex]
2. Calculate the Total Alarm Clock Purchases:
We sum the purchases for just alarm clocks.
[tex]\[
\begin{aligned}
\text{Alarm Clock Purchases} &= \text{Remodel Alarm Clock} + \text{Repair Alarm Clock} + \text{New Purchase Alarm Clock} \\
&= 83 + 41 + 17 \\
&= 141
\end{aligned}
\][/tex]
3. Calculate the Total New Purchases:
We sum the new purchase entries across all types.
[tex]\[
\begin{aligned}
\text{New Purchases Total} &= \text{New Purchase Watch} + \text{New Purchase Clock} + \text{New Purchase Alarm Clock} \\
&= 19 + 11 + 17 \\
&= 47
\end{aligned}
\][/tex]
4. Calculate the Overlap of New Purchases and Alarm Clock Purchases:
The overlap refers to the customers who have purchased a new alarm clock, which is already counted in both totals above.
[tex]\[
\text{Overlap (New Purchase Alarm Clock)} = 17
\][/tex]
5. Calculate the Probability using the Formula for [tex]\(P(A \text{ or } B)\)[/tex]:
The formula for the probability of A or B occurring is:
[tex]\[
P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)
\][/tex]
For our problem:
[tex]\[
P(\text{Alarm Clock or New Purchase}) = \frac{\text{Alarm Clock Purchases} + \text{New Purchases Total} - \text{Overlap}}{\text{Total Purchases}}
\][/tex]
[tex]\[
\begin{aligned}
P(\text{Alarm Clock or New Purchase}) &= \frac{141 + 47 - 17}{411} \\
&= \frac{171}{411} \\
&= 0.41605839416058393
\end{aligned}
\][/tex]
So, the probability that a randomly chosen customer has purchased an alarm clock or a new purchase is approximately 0.416 in simplest form.
[tex]\[
\boxed{0.416}
\][/tex]
