A surprising invention

Since the machine has a one in four chance of cleaning the dishes, and the machine does each chore for 5 minutes, the machine is likely to clean the dishes for five minutes every 20 minutes. Since there are three "20 minutes" in an hour, we can expect the machine to clean the dishes 3 times a day, or 15 minutes an hour

As per expected value wiki . This solution offers a more formal view of the problem.

Let $X_i$ mark the number of times the robot cleans the dishes in an interval $i$ . Then we have the expected value of $X_i$ :

$E\left[X_i\right] = \frac{1}{4} \cdot + \frac{1}{3} \cdot 0 + \frac{5}{12} \cdot 0 = \frac{1}{4}$

Because there is $\frac{60}{5}=12$ intervals in an hour and $E\left[X_i\right]$ is the same for all $i$ , we can then write:

$E\left[X\right] = E\left[\sum_{i=1}^{12} X_i \right] = \sum_{i=1}^{12} E\left[X_i\right] = 12 \cdot E\left[X_i\right] = 12 \cdot \frac{1}{4} = \boxed{3}$