# Video: Finding the Union Probability of Two Events and Its Complementary Probability in a Real-World Problem

In an animal rescue shelter, 39% of the current inhabitants are cats (C) and 41% are dogs (D). Find the probability that an animal chosen at random is either a cat or a dog. Find the probability that an animal chosen at random is neither a cat nor a dog.

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### Video Transcript

In an animal rescue shelter, 39 percent of the current inhabitants are cats, C, and 41 percent are dogs, D. Find the probability that an animal chosen at random is either a cat or a dog. Find the probability that an animal chosen at random is neither a cat nor a dog.

Let’s see what we know. In an animal rescue shelter, 39 percent of the animals are cats and 41 percent of the animals are dogs. For problem one, we wanna find the probability that an animal chosen at random is either a cat or a dog.

We know that animals can either be a cat or a dog, but not both, which makes these mutually exclusive events. And that means, to find the probability of event C or D happening, we add the probability of event C and the probability of event D. Remember, the probability of event A occurring is the number of ways A can occur over all possible outcomes. Probability of C would be the probability a cat is chosen out of all possible animals.

If 39 percent of all the animals are cats, then there is a probability of 0.39 that a cat would be chosen out of all possible animals. In the same way, if 41 percent of all the animals are dogs, then the probability of choosing a dog out of all possible animals is 0.41. If we combine those two probabilities, 0.39 plus 0.41, we see that the probability of selecting a cat or a dog, if chosen randomly, is 0.80. That also means we know that 80 percent of the animals in the shelter are either a cat or a dog. If 80 percent of the animals are either cats or dogs, then 20 percent of the animals are not cats or dogs.

To find the probability that an animal chosen at random is neither a cat nor a dog, we could find the compliment of the probability that it is a cat or a dog. Since the compliment is everything that is not C or D. And we find that by taking the probability of C or D and subtracting that from one. The probability that you will not select a cat or a dog is then 0.20.