Final Answer:
The probability of owning a cat is 0.48.
The probability of owning any of the three types of dogs is 0.39.
The probability of owning a cat and any dog is 0.25.
The probability of someone owning a cat given that they own any dog is 0.641.
The probability of someone owning any dog given that they own a cat is 0.52.
Step-by-step explanation:
The probability of owning a cat can be found by summing the probabilities in the "Cat" row of the contingency table. Therefore, P(Cat) = P(Cat and No Dog) + P(Cat and Golden Doodle) + P(Cat and Corgi) + P(Cat and Husky) = 0.3 + 0.1 + 0.05 + 0.03 = 0.48.
To determine the probability of owning any of the three types of dogs, we sum the probabilities in the "Golden Doodle," "Corgi," and "Husky" columns. Therefore, P(Any Dog) = P(No Cat and Golden Doodle) + P(Cat and Golden Doodle) + P(No Cat and Corgi) + P(Cat and Corgi) + P(No Cat and Husky) + P(Cat and Husky) = 0.12 + 0.06 + 0.23 + 0.11 = 0.39.
The probability of owning a cat and any dog is obtained by summing the probabilities in the cells where both a cat and a dog are present. Therefore, P(Cat and Any Dog) = P(Cat and Golden Doodle) + P(Cat and Corgi) + P(Cat and Husky) = 0.1 + 0.05 + 0.03 = 0.25.
To find the probability of someone owning a cat given that they own any dog, we use the conditional probability formula: P(Cat | Any Dog) = P(Cat and Any Dog) / P(Any Dog) = 0.25 / 0.39 ≈ 0.641.
Finally, the probability of someone owning any dog given that they own a cat is determined using the conditional probability formula: P(Any Dog | Cat) = P(Cat and Any Dog) / P(Cat) = 0.25 / 0.48 ≈ 0.52.