Questions: A company that makes dog treats is investigating whether a dog's size affects if the dog enjoys the treat or not. The company asked 135 dog owners if their dog enjoyed the chicken-flavored chew sticks. Their responses are given in the following table. Dog Size Enjoyed Did Not Enjoy Small 11 20 Medium 16 28 Large 50 10 If one response is selected at random, what is the probability the response indicated that the dog is small-sized given that they enjoyed the treat? Express your answer as a fraction in lowest terms or a decimal rounded to the nearest millionth.

A company that makes dog treats is investigating whether a dog's size affects if the dog enjoys the treat or not. The company asked 135 dog owners if their dog enjoyed the chicken-flavored chew sticks. Their responses are given in the following table. Dog Size Enjoyed Did Not Enjoy Small 11 20 Medium 16 28 Large 50 10 If one response is selected at random, what is the probability the response indicated that the dog is small-sized given that they enjoyed the treat? Express your answer as a fraction in lowest terms or a decimal rounded to the nearest millionth.
Transcript text: A company that makes dog treats is investigating whether a dog's size affects if the dog enjoys the treat or not. The company asked 135 dog owners if their dog enjoyed the chicken-flavored chew sticks. Their responses are given in the following table. \begin{tabular}{|c|c|c|} \hline Dog Size & Enjoyed & Did Not Enjoy \\ \hline Small & 11 & 20 \\ \hline Medium & 16 & 28 \\ \hline Large & 50 & 10 \\ \hline \end{tabular} If one response is selected at random, what is the probability the response indicated that the dog is small-sized given that they enjoyed the treat? Express your answer as a fraction in lowest terms or a decimal rounded to the nearest millionth.
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Solution

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Solution Steps

Step 1: Calculate Total Enjoyed

To find the total number of dogs that enjoyed the treat, we sum the number of dogs that enjoyed the treat across all sizes:

\[ \text{Total Enjoyed} = 11 + 16 + 50 = 77 \]

Step 2: Identify Small Dogs that Enjoyed

Next, we identify the number of small dogs that enjoyed the treat, which is given as:

\[ \text{Small Enjoyed} = 11 \]

Step 3: Calculate Probability

We now calculate the probability that a dog is small-sized given that they enjoyed the treat using the formula:

\[ P(\text{Small} \mid \text{Enjoyed}) = \frac{\text{Small Enjoyed}}{\text{Total Enjoyed}} = \frac{11}{77} \]

This fraction simplifies to:

\[ P(\text{Small} \mid \text{Enjoyed}) = \frac{1}{7} \]

Final Answer

The probability that a dog is small-sized given that they enjoyed the treat is

\[ \boxed{\frac{1}{7}} \]

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