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:

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").