A vacuum cleaning robot is equipped with a cleaning unit to clean the floor. Furthermore, the robot has a sensor to detect whether the floor is clean or dirty. Neither the cleaning unit nor the sensor are perfect. From previous experience you know that the robot succeeds in cleaning a dirty floor with a probability of
$p(x_{t+1} = clean|x_t = dirty, u_{t+1} = vacuum - clean) = 0.7$
where $x_{t+1}$ is the state of the floor after having vacuum-cleaned, $u_{t+1}$ is the control command, and $x_t$ is the state of the floor before performing the action.
The probability that the sensor indicates that the floor is clean although it is dirty is given by $p(z = clean | x = dirty) = 0.3$, and the probability that the sensor correctly detects a clean floor is given by $p(z = clean | x = clean) = 0.9$.
Unfortunately, you have no knowledge about the current state of the floor. However, after cleaning the floor the sensor of the robot indicates that the floor is clean.
a) Compute the probability that the floor is still dirty after the robot have vacuum-cleaned Use an appropriate prior distribution and justify your choice.
b) Which prior gives you a lower bound for that probability?