Question

In a certain community, 30% of the families own a dog, and 20% of the families that own a dog also own a cat. It is also known that 34% of all the families own a cat. What is the probability that a randomly selected family owns a cat? What is the conditional probability that a randomly selected family doesn't own a dog given that it owns a cat?

Answer 1

the answer is 58.8%

Explanation:

From the example given, we apply the use of conditional probability

The general rule for conditional probability is: P(A given B) = Probability of (A and B) / Probability (B).

The Probability of a randomly select family owns a cat is 34%.

The Probability of not owing a dog (A), given that it owns a cat (B) = P(not owing a dog AND owning a cat) / P (owning a cat). which is = x/y so

The relationship between a Cat and Dog is not determined to be independent, thus, we say x just by finding the probability of the two events and multiplying.

Given that 30% own a dog and 20% of those own a cat is 30% x 20% = 6%

7% of all the families own both.

Since 34% own a cat, 34 - 6 = 28% own a cat but not a dog. That gives you x, and y is still just 34%,

So for x/y = 20/34

= 58.8%

Answer 2

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%