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Answer:
What is the probability that a randomly selected family owns a cat? 34%
What is the conditional probability that a randomly selected family doesn't own a dog given that it owns a cat? 82.4%
Step-by-step explanation: We can use a Venn (attached) diagram to describe this situation:
Imagine a community of 100 families (we can assum a number, because in the end, it does not matter)
So, 30% of the families own a dog = .30*100 = 30
20% of the families that own a dog also own a cat = 0.2*30 = 6
34% of all the families own a cat = 0.34*100 = 34
Dogs and cats: 6
Only dogs: 30 - 6 = 24
Only cats: 34 - 6 = 28
Not cat and dogs: 24+6+28 = 58; 100 - 58 = 42
What is the probability that a randomly selected family owns a cat?
34/100 = 34%
What is the conditional probability that a randomly selected family doesn't own a dog given that it owns a cat?
A = doesn't own a dog
B = owns a cat
P(A|B) = P(A∩B)/P(B) = 28/34 = 82.4%
For part (a), you have
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For part (b), since the degrees of the numerator and denominator are the same, you first need to find the quotient and remainder upon division.
In the remainder term, the denominator
can't be factorized into linear components with real coefficients, since the discriminant is negative 
. However, you can still factorized over the complex numbers, so a partial fraction decomposition in terms of complexes does exist.
Then you have
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So, you could write
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If
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So,
For part (b), since the degrees of the numerator and denominator are the same, you first need to find the quotient and remainder upon division.
In the remainder term, the denominator
Then you have
When
When
So, you could write
but that may or may not be considered acceptable by that webpage.