Final answer:
Statistical independence between being a cat and being adopted is shown when the fraction of cats adopted is the same as that not adopted, revealing that knowing an animal is a cat doesn't affect the probability of its adoption.
Step-by-step explanation:
The fact that establishes that the observation "an animal is a cat" is independent from the observation "an animal is adopted" is that cats constitute the same fraction of animals adopted as not adopted. This indicates that the likelihood of a cat being adopted is the same regardless of whether we know it is a cat or not, which demonstrates statistical independence.
Independence means that the outcome of one event does not affect the outcome of another. In the context of probability, two events are independent if the occurrence of one does not change the probability of occurrence of the other. If the probability of adopting a cat is the same as the probability of not adopting a cat, it implies that being a cat is not a factor that affects the adoption rate.