Question

The local animal shelter in Exercise 8 reported that it currently has 24 dogs and 18 cats available for adoption; 8 of the dogs and 6 of the cats are male. Are the species and sex of the animals independent? Explain.

Short answer

The species and sex of the animals are independent because the probabilities match.

Step-by-step solution

Understand Independence

Two categorical events, such as species and sex in this problem, are independent if the probability of their intersection is the product of their individual probabilities. Specifically, they are independent if \( P(A \cap B) = P(A) \times P(B) \), where \(A\) and \(B\) are the events.

Define Events and Probabilities

Let \( A \) be the event of selecting a male animal, and \( B \) be the event of selecting a dog. We know there are 42 animals total (dogs and cats combined). The probability of selecting a male animal, \( P(A) \), is the total number of male animals (8 dogs + 6 cats = 14 males) divided by the total number of animals: \( P(A) = \frac{14}{42} = \frac{1}{3} \). The probability of selecting a dog, \( P(B) \), is the number of dogs (24) divided by the total number of animals: \( P(B) = \frac{24}{42} = \frac{4}{7} \).

Calculate Intersection Probability

Next, calculate the probability of selecting an animal that is both a male and a dog, \( P(A \cap B) \). There are 8 male dogs, so \( P(A \cap B) = \frac{8}{42} = \frac{4}{21} \).

Check Independence Criterion

Now, check if the independence criterion holds: \( P(A \cap B) = P(A) \times P(B) \). Substitute the probabilities: \( \frac{4}{21} = \frac{1}{3} \times \frac{4}{7} = \frac{4}{21} \). Since both sides of the equation are equal, we confirm that the species and sex of the animals are independent.

Key concepts

Probability

Probability is a fascinating concept that helps us understand the likelihood of various events happening. When discussing probability, it's often expressed as a ratio or fraction. For instance, if there's a one in three chance of picking a specific outcome, the probability would be represented as \( \frac{1}{3} \).

Let's break down the calculation of probability with an example. Suppose you have 42 animals in a shelter, consisting of both dogs and cats. If you want to know the probability of picking a male animal at random, you would take the number of male animals and divide it by the total number of animals. With 14 male animals in the group, the probability would be \( \frac{14}{42} \), which simplifies to \( \frac{1}{3} \).

Probability is not just a pattern but an essential tool in statistics that underlies many real-world decisions and predictions. From predicting weather patterns to designing experiments, understanding probability can be a powerful advantage.

Categorical Variables

In statistics, categorical variables are those that represent distinct categories or groups. Unlike continuous variables that can take on an infinite range of values, categorical variables are more about classifications.

For example, in our exercise, you have two primary categories of data: species (dogs and cats) and sex (male and female). These are categorical because each animal can be classified distinctly within these groups.

Categorical variables are crucial when conducting analyses that involve distinguishing between different segments of data. They're often used in surveys and studies where characteristics like gender, types of animals, or even types of industries are surveyed. Understanding how these categories interact, as in our scenario with animal breeds and gender, enables meaningful insights, especially when assessing probabilities or dependencies.

Intersection of Events

The intersection of events in probability refers to situations where two or more events occur at the same time. This is a fundamental concept when you want to determine scenarios where conditions overlap.

In the context of our exercise, the intersection of events involves determining the number of male dogs at the shelter. The event of choosing a male dog is the intersection of the event "choosing a male" and the event "choosing a dog." To find this intersection probability, you look at how many dogs are also male. In this case, there are eight male dogs, so the probability of this intersection, expressed as \( P(A \cap B) \), is \( \frac{8}{42} \).

Understanding event intersections helps in calculating the likelihood of combined scenarios, forming the backbone of many statistical analyses concerning joint probabilities.