https://file.aiquickdraw.com/mathbot/file/1727059091835mlkrsgqh.jpg

Let's break down the problem shown in the image step by step:

The table shows preferences for various pets based on age. The table looks like this:

|               | Cats | Dogs | Other | Total |
|---------------|------|------|-------|-------|
| **Over 40**   | 100  | 50   | 30    | ?     |
| **40 or Under** | 50   | 50   | 20    | ?     |
| **Total**     | ?    | ?    | ?     | ?     |

The problem asks us to find two conditional probabilities:

### Part (a) 
Find the probability that a person **owns a dog**, given that they are **"40 or Under"**.

#### Solution:
This is a conditional probability problem where we use the formula:
\[
P(\text{Dog | 40 or Under}) = \frac{P(\text{Dog and 40 or Under})}{P(\text{40 or Under})}
\]

- **P(Dog and 40 or Under):** From the table, the number of people 40 or under who own a dog is **50**.
- **P(40 or Under):** The total number of people who are 40 or under is the sum of all preferences in the "40 or Under" row: \(50 + 50 + 20 = 120\).

Thus, the probability is:
\[
P(\text{Dog | 40 or Under}) = \frac{50}{120}
\]

### Part (b)
Find the probability that a person is **"40 or Under"**, given that they **own a dog**.

#### Solution:
This is another conditional probability where we use the formula:
\[
P(\text{40 or Under | Dog}) = \frac{P(\text{40 or Under and Dog})}{P(\text{Dog})}
\]

- **P(40 or Under and Dog):** Again, from the table, the number of people 40 or under who own a dog is **50**.
- **P(Dog):** The total number of people who own a dog is the sum of the "Dogs" column: \(50 + 50 = 100\).

Thus, the probability is:
\[
P(\text{40 or Under | Dog}) = \frac{50}{100}
\]

### Final Answers:
- **a)** \(\frac{50}{120}\)
- **b)** \(\frac{50}{100}\)

Do you want further details or have any questions?

Here are 5 related questions:
1. How would you calculate the probability of owning a cat given that the person is over 40?
2. What is the total number of people surveyed based on the table?
3. How does the conditional probability formula work in general cases?
4. Can you find the probability of being over 40 given that the person owns a pet in the "Other" category?
5. What is the difference between joint probability and conditional probability?

**Tip:** When solving conditional probability problems, always make sure you correctly identify both the numerator (joint probability) and the denominator (the condition being "given").

A person is selected at random. Find the probability that they: a) Own a dog, given that they are '40 or Under'. b) Are '40 or Under', given that they own a dog.

Solve a conditional probability problem involving dog ownership and age. Learn how to calculate the probability of owning a dog given that the person is 40 or under, and the probability of being 40 or under given that the person owns a dog. Suitable for high school students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation