Sure, let's solve this step-by-step by using the information provided in the survey data table.
1. Understanding the question:
- We are asked to find the likelihood that a student does not have a pet given that they have a sibling.
2. Key definitions from probability:
- The likelihood that a student does not have a pet given that they have a sibling can be found by using conditional probability.
- This is mathematically represented as [tex]\( P(\text{No Pet} \mid \text{Sibling}) \)[/tex].
3. Reading the table:
- The table provides the following data:
- [tex]\( P(\text{Sibling and Pet}) = 0.3 \)[/tex]
- [tex]\( P(\text{Sibling and No Pet}) = 0.45 \)[/tex]
- Total proportion of students with siblings ([tex]\( P(\text{Sibling}) \)[/tex]) = 0.75
4. Conditional Probability Formula:
- The conditional probability [tex]\( P(A \mid B) \)[/tex] is given by:
[tex]\[
P(A \mid B) = \frac{P(A \cap B)}{P(B)}
\][/tex]
- Here [tex]\( A \)[/tex] is the event of not having a pet, and [tex]\( B \)[/tex] is the event of having a sibling.
5. Applying the formula:
- [tex]\( P(\text{No Pet} \mid \text{Sibling}) = \frac{P(\text{No Pet and Sibling})}{P(\text{Sibling})} \)[/tex]
6. Plugging in the numbers:
- [tex]\( P(\text{No Pet and Sibling}) = 0.45 \)[/tex]
- [tex]\( P(\text{Sibling}) = 0.75 \)[/tex]
- Therefore:
[tex]\[
P(\text{No Pet} \mid \text{Sibling}) = \frac{0.45}{0.75}
\][/tex]
7. Simplification:
- Simplifying the fraction:
[tex]\[
P(\text{No Pet} \mid \text{Sibling}) = 0.6
\][/tex]
8. Convert to percentage:
- The likelihood as a percentage is:
[tex]\[
0.6 \times 100 = 60\%
\][/tex]
9. Conclusion:
- Given that a student has a sibling, the likelihood that they do not have a pet is [tex]\( 60\% \)[/tex].
Therefore, the correct answer is:
C. 60%
