To determine the probability that a randomly chosen customer has purchased an alarm clock or made a new purchase, we need to follow several steps:
1. Calculate the total number of customers:
We will sum up all the given values for each category:
- Watch: [tex]\(73 + 47 + 19 = 139\)[/tex]
- Clock: [tex]\(61 + 59 + 11 = 131\)[/tex]
- Alarm Clock: [tex]\(83 + 41 + 17 = 141\)[/tex]
So, the total number of customers is:
[tex]\[
139 + 131 + 141 = 411
\][/tex]
2. Calculate the total number of customers who purchased an alarm clock:
Customers who purchased an alarm clock fall into all three categories (Remodel, Repair, New Purchase):
[tex]\[
83 + 41 + 17 = 141
\][/tex]
3. Calculate the total number of customers who made a new purchase:
Customers who made a new purchase fall into each type of clock (Watch, Clock, Alarm Clock):
[tex]\[
19 + 11 + 17 = 47
\][/tex]
4. Calculate the overlap between customers who made a new purchase and those who purchased an alarm clock:
Specifically, these are the customers who made a new purchase of an alarm clock, which is:
[tex]\[
17
\][/tex]
5. Calculate the total number of customers who purchased an alarm clock or made a new purchase:
Since the customers who made a new purchase of an alarm clock are counted in both categories, we need to subtract this overlap once:
[tex]\[
141 + 47 - 17 = 171
\][/tex]
6. Calculate the probability:
The number of favorable outcomes (customers who either purchased an alarm clock or made a new purchase) is [tex]\(171\)[/tex], out of a total of [tex]\(411\)[/tex] customers.
Therefore, the probability is:
[tex]\[
P(\text{Alarm Clock or New Purchase}) = \frac{171}{411}
\][/tex]
7. Simplify the fraction to its simplest form:
Both [tex]\(171\)[/tex] and [tex]\(411\)[/tex] can be simplified by finding their greatest common divisor (GCD). The GCD of [tex]\(171\)[/tex] and [tex]\(411\)[/tex] is [tex]\(3\)[/tex]. So, we simplify the fraction:
[tex]\[
\frac{171 \div 3}{411 \div 3} = \frac{57}{137}
\][/tex]
Thus, the probability that a randomly chosen customer has purchased an alarm clock or made a new purchase is:
[tex]\[
P(\text{Alarm Clock or New Purchase}) = \frac{57}{137}
\][/tex]
