Question

in a class of students, the following data table summarizes how many students have a cat or a dog. what is the probability that a student has a dog given that they do not have a cat?

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$$\begin{tabular}{|c|c|c|} \\hline & has a cat & does not have a cat \\\\ \\hline has a dog & 3 & 10 \\\\ \\hline does not have a dog & 6 & 2 \\\\ \\hline \\end{tabular}$$

Explanation:

Identify the given condition

Using the Two-Way Tables knowledge point, we first identify the subset of students who satisfy the given condition: "does not have a cat". Looking at the column labeled "Does not have a cat", we find the total number of students in this category by adding the values in that column:

$$ \text{Total without a cat} = 10 + 2 = 12 $$

Identify the favorable outcomes

Next, we find the number of students within this restricted group who also satisfy the target event: "has a dog". From the "Does not have a cat" column, the number of students who "Has a dog" is:

$$ \text{Favorable outcomes} = 10 $$

Calculate the conditional probability

We calculate the probability by dividing the number of favorable outcomes by the total number of outcomes in our restricted group:

$$ P(\text{Dog} \mid \text{No Cat}) = \frac{10}{12} = \frac{5}{6} $$

Answer:

\(\frac{5}{6}\)