$\times 1$ 2 $\times 3$ 5 6. 7 8 2. Dr. Howard is a veterinarian who sees only dogs and cats. In each appointment, she may or may not give the animal a vaccine. The two-way frequency table summarizes Dr. Howard"s 50 appointments last week. \begin{tabular}{|c|c|c|} \cline { 2 - 3 } \multicolumn{1}{c|}{} & Dog & Cat \\ \hline Vaccine & 12 & 18 \\ \hline No vacdine & 13 & 7 \\ \hline \end{tabular} 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, Write your answers as decimals. (a) $P($ vaccine $)=[$ (b) P(dog and vaccine) $=\square$ (c) P(dog I vaccine $)=\square$ Check

Step 1 ：Given that Dr. Howard had 50 appointments last week, and the two-way frequency table summarizes the appointments.

Step 2 ：From the table, we can see that there were 30 appointments where a vaccine was given (12 to dogs and 18 to cats).

Step 3 ：There were also 12 appointments where a dog was given a vaccine.

Step 4 ：We can calculate the probability of each event as follows:

Step 5 ：(a) The probability that a randomly chosen appointment included a vaccine, denoted as \(P(\text{vaccine})\), is the number of appointments that included a vaccine divided by the total number of appointments. So, \(P(\text{vaccine}) = \frac{30}{50} = 0.6\).

Step 6 ：(b) The probability that a randomly chosen appointment involved a dog and a vaccine, denoted as \(P(\text{dog and vaccine})\), is the number of appointments that involved a dog and a vaccine divided by the total number of appointments. So, \(P(\text{dog and vaccine}) = \frac{12}{50} = 0.24\).

Step 7 ：(c) The conditional probability that a randomly chosen appointment involved a dog given that it included a vaccine, denoted as \(P(\text{dog | vaccine})\), is the number of appointments that involved a dog and a vaccine divided by the number of appointments that included a vaccine. So, \(P(\text{dog | vaccine}) = \frac{12}{30} = 0.4\).

Step 8 ：Final Answer: \(\boxed{(a) P(\text{vaccine}) = 0.6, (b) P(\text{dog and vaccine}) = 0.24, (c) P(\text{dog | vaccine}) = 0.4}\)