Question

In a certain community, 25% of the families own a dog, and 20% of the families that own a dog also own a cat. It is also known that 35% of all the families own a cat.
What is the probability that a randomly selected family owns a cat? 0.55
What is the conditional probability that a randomly selected family owns a dog given that it doesn't own a cat?

Answer 1

P(cat) = 0.35 {given}

P(both dog and cat) = 0.25*0.20 = 0.05

therefore, P(not dog, not cat) = 1 - (P(cat) + P(dog) - P(both dog and cat) )

= 1 - (0.35+0.25-0.05)

= 0.45

P(not cat) = P(not cat | dog)*P(dog) + P(not cat | not dog)*P(not dog)

= ( (P(dog) - P(both))/P(dog) )*P(dog) + ( P(not cat, not dog)/P(not dog) )*P(not dog)

= ( (0.25-0.05)/0.25)*0.25 + (0.45/(1-0.25))*(1-0.25)

= 0.20 + 0.45

= 0.65

P(dog | not cat) = P(not cat | dog)*P(dog) / P(not cat)

= 0.20 / 0.65

= 0.3077

P(dog | not cat) = 0.3077

(please UPVOTE)