Professor Guilherme created three strange machines. Machine $A$ transforms a cat into a dog with a probability of $\frac{1}{3}$. Machine $B$ transforms a cat into a dog with a probability of $\frac{2}{5}$. Machine $C$ transforms a cat into a dog with a probability of $\frac{1}{4}$. And if the animal is a dog, none of the machines will perform any transformation.

Professor Guilherme placed a cat in machine $A$, then placed the resulting animal from machine $A$ in machine $B$, and finally placed the resulting animal from machine $B$ in machine $C$. What is the probability that a dog came out of machine $C$?
Solution

First, let's calculate the probability of a cat coming out as a cat from each machine.

Since the probability of a dog coming out of machine $A$ is $\frac{1}{3}$, the probability of a cat coming out of this machine is $1-\frac{1}{3}=\frac{2}{3}$.

Since the probability of a dog coming out of machine $B$ is $\frac{2}{5}$, the probability of a cat coming out of this machine is $1-\frac{2}{5}=\frac{3}{5}$.

Since the probability of a dog coming out of machine $C$ is $\frac{1}{4}$, the probability of a cat coming out of this machine is $1-\frac{1}{4}=\frac{3}{4}$.

For a cat to come out as a cat after passing through the three machines, it is necessary that it comes out as a cat from each of them, because once it becomes a dog, no machine will transform it back into a cat. Therefore, the probability of the cat coming out as a cat after passing through the three machines is the product of the three probabilities:

$$
\frac{2}{3} \times \frac{3}{5} \times \frac{3}{4}=\frac{3}{10}
$$

The probability that this cat will come out as a dog after passing through the three machines is one minus the probability that it will come out as a cat after passing through the three machines. Since

$$
1-\frac{3}{10}=\frac{7}{10}
$$

the final answer is $\frac{7}{10}$.