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One day at the dog park, John counted 135 dogs, of which 47 were off-leash playing. Of the 47, 10 were Retrievers. He saw a total of 35 Retrievers. Using this information, complete the contingency table below. Then use it to find the probabilities. Reduce all fractions and use / for the fraction bar.

\begin{tabular}{|l|l|l|l|}
\hline & Not on a Leash & Leashed & Total \\
\hline Retriever & 10 & & 35 \\
\hline Not a Retriever & & & \\
\hline Total & 47 & & 135 \\
\hline
\end{tabular}



Answer :

GinnyAnswer
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]

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