To solve for the probability that a randomly chosen customer has either purchased an alarm clock or made a new purchase, we can follow these steps:
1. Calculate the Total Purchases for Each Category:
- Watches:
- Remodel: 73
- Repair: 47
- New Purchase: 19
- Clocks:
- Remodel: 61
- Repair: 59
- New Purchase: 11
- Alarm Clocks:
- Remodel: 83
- Repair: 41
- New Purchase: 17
2. Find the Total Purchases for Alarm Clocks:
[tex]\[
\text{Total Alarm Clock Purchases} = 83 + 41 + 17 = 141
\][/tex]
3. Find the Total Purchases for New Purchases Regardless of Category:
[tex]\[
\text{Total New Purchases} = 19 + 11 + 17 = 47
\][/tex]
4. Find the Total Purchases Overall:
[tex]\[
\text{Total Purchases} = (73 + 47 + 19) + (61 + 59 + 11) + (83 + 41 + 17) = 411
\][/tex]
5. Use the Principle of Inclusion and Exclusion:
We need to add the number of Alarm Clock purchases to the number of New Purchases and then subtract the number of instances where a purchase falls into both categories (i.e., new purchases of alarm clocks), to avoid double-counting those:
[tex]\[
\text{Purchases of Alarm Clocks or New Purchases} = 141 + 47 - 17 = 171
\][/tex]
6. Calculate the Probability:
The probability is then the number of purchases of either an alarm clock or a new purchase divided by the total number of purchases:
[tex]\[
P(\text{Alarm Clock or New Purchase}) = \frac{171}{411}
\][/tex]
7. Simplify the Fraction:
Simplify the fraction to the greatest extent possible. However, as calculated:
[tex]\[
\frac{171}{411} \approx 0.41605839416058393
\][/tex]
So, the probability that a randomly chosen customer has purchased an alarm clock or made a new purchase is approximately [tex]\( 0.416 \)[/tex].
