Let's complete the contingency table and then use it to find the probabilities.
1. Determine the number of off-leash non-retrievers:
- Total off-leash dogs: 47
- Off-leash retrievers: 10
- Off-leash non-retrievers: 47 - 10 = 37
2. Determine the number of leashed retrievers:
- Total retrievers: 35
- Off-leash retrievers: 10
- Leashed retrievers: 35 - 10 = 25
3. Determine the number of leashed non-retrievers:
- Total dogs: 135
- Total retrievers: 35
- Total non-retrievers: 135 - 35 = 100
- Off-leash non-retrievers: 37
- Leashed non-retrievers: 100 - 37 = 63
4. Fill in the contingency table:
[tex]\[
\begin{tabular}{|l|l|l|l|}
\hline & Not on a Leash & Leashed & Total \\
\hline Retriever & 10 & 25 & 35 \\
\hline Not a Retriever & 37 & 63 & 100 \\
\hline Total & 47 & 88 & 135 \\
\hline
\end{tabular}
\][/tex]
5. Calculate probabilities:
- Probability that a random dog is a retriever and not on a leash:
[tex]\[
\frac{\text{Off-leash retrievers}}{\text{Total dogs}} = \frac{10}{135} = 0.07407407407407407
\][/tex]
- Probability that a random dog is not a retriever and not on a leash:
[tex]\[
\frac{\text{Off-leash non-retrievers}}{\text{Total dogs}} = \frac{37}{135} = 0.2740740740740741
\][/tex]
- Probability that a random dog is a retriever and leashed:
[tex]\[
\frac{\text{Leashed retrievers}}{\text{Total dogs}} = \frac{25}{135} = 0.18518518518518517
\][/tex]
- Probability that a random dog is not a retriever and leashed:
[tex]\[
\frac{\text{Leashed non-retrievers}}{\text{Total dogs}} = \frac{63}{135} = 0.4666666666666667
\][/tex]
Summary:
[tex]\[
\begin{aligned}
&\text{Probability of a retriever not on a leash: } 0.07407407407407407 \\
&\text{Probability of not a retriever not on a leash: } 0.2740740740740741 \\
&\text{Probability of a retriever leashed: } 0.18518518518518517 \\
&\text{Probability of not a retriever leashed: } 0.4666666666666667
\end{aligned}
\][/tex]
