Question

dr. long 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. longs 60 appointments last week.

	vaccine	no vaccine
cat	18	12

let cat be the event that a randomly chosen appointment (from the table) involved a cat.
let no vaccine be the event that a randomly chosen appointment (from the table) did not include a vaccine.
find the following probabilities. write your answers as decimals.
(a) ( p(\text{cat}) = )
(b) ( p(\text{no vaccine and cat}) = )
(c) ( p(\text{no vaccine} mid \text{cat}) = )

Explanation:

Step1: Calculate total appointments

Total appointments = 8 + 22 + 18 + 12 = 60.

Step2: Find P(cat)

Number of cat appointments = 18 + 12 = 30.
$P(\text{cat}) = \frac{\text{Number of cat appointments}}{\text{Total appointments}} = \frac{30}{60} = 0.5$.

Step3: Find P(no vaccine and cat)

Number of cat appointments with no vaccine = 12.
$P(\text{no vaccine and cat}) = \frac{\text{Cat, no vaccine}}{\text{Total}} = \frac{12}{60} = 0.2$.

Step4: Find P(no vaccine | cat)

Using conditional probability formula: $P(A|B) = \frac{P(A \cap B)}{P(B)}$.
Here, $A$ is no vaccine, $B$ is cat.
$P(\text{no vaccine} | \text{cat}) = \frac{\text{Cat, no vaccine}}{\text{Number of cat appointments}} = \frac{12}{30} = 0.4$.

Answer:

s:
(a) $\boldsymbol{0.5}$
(b) $\boldsymbol{0.2}$
(c) $\boldsymbol{0.4}$