Question

One household is to be selected at random from a town.

The probability that the household has a cat is 20%
The probability that the household has a dog is 30%
The probability that the household has a cat or a dog is 45%
What is the probability that the household has a dog, given that the household has a cat?

Answer 1

The probability that the household has a dog given that the household has a cat is 37.5%

How to calculate?

The probability that the household has a dog given that the household has a cat can be found using Bayes' Theorem.

Applying that, we have P(dog | cat) = P(cat and dog) / P(cat) = (P(cat) * P(dog | cat)) / P(cat) = (0.20 * P(dog | cat)) / 0.20 = P(dog | cat)

P(cat or dog) = P(cat) + P(dog) - P(cat and dog) = 0.45

P(cat and dog) = P(cat) * P(dog | cat) = 0.20 * P(dog | cat)

P(dog) = P(dog or cat) - P(cat) = 0.30

P(cat and dog) = P(dog or cat) - (P(dog) - P(cat and dog)) = 0.45 - (0.30 - P(cat and dog))

P(cat and dog) = 0.45 - 0.30 + P(cat and dog)

P(cat and dog) = 0.15 + P(cat and dog)

2 * P(cat and dog) = 0.15 + 2 * P(cat and dog)

P(cat and dog) = 0.15 / 2

P(dog | cat) = P(cat and dog) / P(cat) = 0.15 / (2 * 0.20) = 0.375

In conclusion, the probability that the household has a dog given that the household has a cat is 37.5%.