A local veterinarian clinic recorded the primary diets of 120 animals (cats and dogs). The results are summarized in the table below. | Animal | Dry food | Wet food | Total | | :--- | :---: | :---: | :---: | | Cats | 35 | 25 | 60 | | Dogs | 45 | 15 | 60 | | Total | 80 | 40 | 120 | If one of these an

A local veterinarian clinic recorded the primary diets of 120 animals (cats and dogs). The… — SAT

A local veterinarian clinic recorded the primary diets of 120 animals (cats and dogs). The results are summarized in the table below.

Animal	Dry food	Wet food	Total
Cats	35	25	60
Dogs	45	15	60
Total	80	40	120

If one of these animals is selected at random, what is the probability that the animal is a dog, given that the animal's primary diet is dry food?

$\frac{9}{16}$ Answer
B
$\frac{3}{8}$
C
$\frac{3}{4}$
D
$\frac{5}{8}$

Answer

The correct answer is the option representing

\frac{9}{16}

The correct answer is

\frac{9}{16}

. To find the probability that the selected animal is a dog given that its primary diet is dry food, we restrict our focus to the animals that eat dry food. According to the table, there are 80 animals in total whose primary diet is dry food. Among these 80 animals, 45 are dogs. Therefore, the probability is

\frac{45}{80}

, which simplifies to

\frac{9}{16}

Step-by-Step Solution

Identify the given condition in the question.

The condition is that the selected animal's primary diet is dry food.

This restricts the sample space to only the animals in the 'Dry food' column.

Find the total number of animals that meet the given condition from the table.

The total number of animals whose primary diet is dry food is 80.

This value serves as the denominator for the conditional probability fraction.

Find the number of dogs within the conditional space identified in the previous step.

The number of dogs that eat dry food is 45.

This value serves as the numerator representing the favorable outcomes.

Calculate the conditional probability by dividing the favorable outcomes by the total outcomes in the restricted sample space.

\frac{45}{80} = \frac{9}{16}

Dividing the numerator by the denominator and simplifying by dividing both by 5 yields the final probability.

Key Concept

Conditional Probability from Two-Way Tables

Alternative Method

Instead of working directly with the counts in the table, you can write the conditional probability formula:

P(\text{Dog} \mid \text{Dry}) = \frac{P(\text{Dog} \cap \text{Dry})}{P(\text{Dry})} = \frac{45/120}{80/120} = \frac{45}{80} = \frac{9}{16}

Estimated Time:50s

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