Questions: Dr. Martin is a veterinarian who sees only dogs and cats. In each appointment, he may or may not give the animal a vaccine. The two-way frequency table summarizes Dr. Martin's 50 appointments last week. Dog Cat --------------------- Vaccine 12 18 No vaccine 13 7 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. (If necessary, consult a list of formulas.) (a) P( vaccine )= (b) P( dog and vaccine )= (c) P( dog vaccine )=

Dr. Martin is a veterinarian who sees only dogs and cats.
In each appointment, he may or may not give the animal a vaccine.
The two-way frequency table summarizes Dr. Martin's 50 appointments last week.

Dog Cat
---------------------
Vaccine 12 18
No vaccine 13 7

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.
(If necessary, consult a list of formulas.)
(a) P( vaccine )=
(b) P( dog and vaccine )=
(c) P( dog vaccine )=

Transcript text: Dr. Martin is a veterinarian who sees only dogs and cats. In each appointment, he may or may not give the animal a vaccine. The two-way frequency table summarizes Dr. Martin's 50 appointments last week. \begin{tabular}{|c|c|c|} \cline { 2 - 3 } \multicolumn{1}{c|}{} & Dog & Cat \\ \hline Vaccine & 12 & 18 \\ \hline No vaccine & 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. (If necessary, consult a list of formulas.) (a) $P($ vaccine $)=$ $\square$ (b) $P($ dog and vaccine $)=$ $\square$ (c) $P($ dog $\mid$ vaccine $)=\square$

Solution

Solution Steps

Step 1: Calculate $ P(\text{vaccine}) $

To find the probability that a randomly chosen appointment included a vaccine, we use the formula:

\[ P(\text{vaccine}) = \frac{\text{Total number of vaccine appointments}}{\text{Total number of appointments}} \]

Given:

Total vaccine appointments = 12 (dogs) + 18 (cats) = 30
Total appointments = 50

\[ P(\text{vaccine}) = \frac{30}{50} = 0.6 \]

Step 2: Calculate $ P(\text{dog and vaccine}) $

To find the probability that a randomly chosen appointment involved a dog and included a vaccine, we use the formula:

\[ P(\text{dog and vaccine}) = \frac{\text{Number of dog appointments with vaccine}}{\text{Total number of appointments}} \]

Given:

Dog appointments with vaccine = 12

\[ P(\text{dog and vaccine}) = \frac{12}{50} = 0.24 \]

Step 3: Calculate $ P(\text{dog} \mid \text{vaccine}) $

To find the probability that a randomly chosen appointment involved a dog given that it included a vaccine, we use the formula:

\[ P(\text{dog} \mid \text{vaccine}) = \frac{\text{Number of dog appointments with vaccine}}{\text{Total number of vaccine appointments}} \]

Given:

Dog appointments with vaccine = 12
Total vaccine appointments = 30

\[ P(\text{dog} \mid \text{vaccine}) = \frac{12}{30} = 0.4 \]

Final Answer

\[ \boxed{ \begin{align_} (a) & \quad P(\text{vaccine}) = 0.6 \\ (b) & \quad P(\text{dog and vaccine}) = 0.24 \\ (c) & \quad P(\text{dog} \mid \text{vaccine}) = 0.4 \\ \end{align_} } \]

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