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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]
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]
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