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Let's analyze the table and the problem from the image you uploaded:

We are given a two-way frequency table of Dr. Long's 50 appointments last week, which involves dogs and cats and whether or not they received a vaccine.

### Table:
|         | Dog | Cat | Total |
|---------|-----|-----|-------|
| Vaccine | 12  | 18  | 30    |
| No Vaccine | 13  | 7   | 20    |
| **Total** | **25**  | **25**  | **50**  |

We need to calculate the following probabilities:

### (a) P(vaccine)
This is the probability that a randomly chosen appointment included a vaccine.
\[
P(\text{vaccine}) = \frac{\text{total appointments with a vaccine}}{\text{total appointments}} = \frac{30}{50} = 0.6
\]

### (b) P(dog and vaccine)
This is the probability that a randomly chosen appointment involved a dog and included a vaccine.
\[
P(\text{dog and vaccine}) = \frac{\text{appointments with dog and vaccine}}{\text{total appointments}} = \frac{12}{50} = 0.24
\]

### (c) P(dog | vaccine)
This is the conditional probability that a randomly chosen appointment involved a dog, given that a vaccine was administered. The formula for conditional probability is:
\[
P(\text{dog | vaccine}) = \frac{P(\text{dog and vaccine})}{P(\text{vaccine})} = \frac{12/50}{30/50} = \frac{12}{30} = 0.4
\]

### Final Answers:
- (a) P(vaccine) = 0.6
- (b) P(dog and vaccine) = 0.24
- (c) P(dog | vaccine) = 0.4

Would you like more details on any part of this? Here are some related questions:

1. What is the probability of having a dog given no vaccine was administered?
2. What is the probability of having a cat and receiving a vaccine?
3. What is the probability of having a dog in total?
4. How would the total probabilities change if more dogs were vaccinated?
5. Can we calculate the probability of getting a cat if we know a vaccine was not given?

**Tip:** Conditional probability helps in narrowing down outcomes based on given conditions, making it essential for real-world decisions like medical treatments or diagnostics.

Let vaccine be the event that a randomly chosen appointment (from the table) included a vaccine. Let dog be the event that a randomly chosen appointment (from the table) involved a dog. Find the following probabilities: (a) P(vaccine), (b) P(dog and vaccine), (c) P(dog | vaccine).

Learn how to calculate probabilities related to vaccines and dogs using a two-way frequency table. This problem covers key probability concepts such as P(vaccine), P(dog and vaccine), and conditional probability P(dog | vaccine). Suitable for Grades 9-12.

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