Question

In its monthly report, the local animal shelter states that it currently has 24 dogs and 18 cats available for adoption. Eight of the dogs and 6 of the cats are male Find each of the following conditional probabilities if an animal is selected at random: a) The pet is male, given that it is a cat. b) The pet is a cat, given that it is female. c) The pet is female, given that it is a dog.

Short answer

a) \( \frac{1}{3} \); b) \( \frac{3}{7} \); c) \( \frac{2}{3} \).

Step-by-step solution

Identify the Total Number of Each Animal and Gender

First, we need to establish the total number of pets, the total number of cats and dogs, and the number of male and female cats and dogs. We have 24 dogs and 18 cats. Of the dogs, 8 are male, so there are 24 - 8 = 16 female dogs. Of the cats, 6 are male, so there are 18 - 6 = 12 female cats.

Calculate Conditional Probability a

For the probability that a pet is male given that it is a cat, we use the formula: \( P( ext{Male} \, | \, ext{Cat}) = \frac{P( ext{Male} \, ext{and} \, ext{Cat})}{P( ext{Cat})} \). We have 6 male cats and 18 total cats, so \( P( ext{Male} \, | \, ext{Cat}) = \frac{6}{18} = \frac{1}{3} \).

Calculate Conditional Probability b

For the probability that the pet is a cat given that it is female, we use: \( P( ext{Cat} \, | \, ext{Female}) = \frac{P( ext{Cat} \, ext{and} \, ext{Female})}{P( ext{Female})} \). We have 12 female cats and a total of 16 + 12 = 28 female animals. Thus, \( P( ext{Cat} \, | \, ext{Female}) = \frac{12}{28} = \frac{3}{7} \).

Calculate Conditional Probability c

For the probability that the pet is female given that it is a dog, we use: \( P( ext{Female} \, | \, ext{Dog}) = \frac{P( ext{Female} \, ext{and} \, ext{Dog})}{P( ext{Dog})} \). We have 16 female dogs and 24 total dogs, thus, \( P( ext{Female} \, | \, ext{Dog}) = \frac{16}{24} = \frac{2}{3} \).

Key concepts

Probability Theory

Conditional probability is a fundamental concept in probability theory. It helps us understand how the likelihood of an event changes when we have additional information. In simple terms, it is the probability of an event happening given that another event has already occurred.

For example, in our exercise, we are finding probabilities based on the condition that an animal is either male or female. The notation for conditional probability is typically shown as \( P(A \mid B) \), which reads as "the probability of A given B". Here, \( A \) and \( B \) are two events. The formula used is:

• \( P(A \mid B) = \frac{P(A \text{ and } B)}{P(B)} \)

This formula shows the probability of both events A and B happening together divided by the probability of B, the event we're conditioning on.

By applying this to different scenarios, like determining the probability of selecting a male cat among all cats, we can make more informed predictions, thus understanding our datasets better.

Statistics Education

Understanding conditional probability is an essential part of statistics education. It is not just about rote computation but also about grasping the concepts to make meaningful analyses.

Students often encounter challenges because probability can be abstract, and datasets can seem too complex. However, exercises like these break down problems into understandable steps. The conditioned event frames probability in everyday contexts, such as determining a characteristic of an animal depending on its gender.

To strengthen your grasp on statistics, remember these key points:

• Practice breaking down problems into smaller, manageable parts.
• Use real-life examples to relate abstract concepts to tangible scenarios.
• Visualize problems using diagrams or charts to see relationships between different data points.

These strategies can help demystify statistics and make it a more approachable and enjoyable subject.

Gender Analysis in Statistics

Gender analysis in statistics involves examining how different variables interact with gender. It plays a crucial role in understanding societal patterns and trends. In our exercise, we look at animals' availability for adoption and explore how gender affects these probabilities.

Such analysis can provide insights into broader gender-related issues. For instance, if a shelter consistently has more male dogs but female cats, a statistical study could explore factors leading to such demographics.

Questions you might consider in gender analysis include:

• What are the underlying reasons for gender imbalances in certain datasets?
• How do gender-based probabilities influence decision-making processes?
• What impact could interventions have to address identified imbalances?

By analyzing statistical data through the lens of gender, decision-makers can tailor approaches more effectively and consciously consider how gender influences outcomes.