Bayes Filter: What is the probability that the object is still blank after the robot has performed an action to paint it?

Question

I know the probability that the robot succeeds in painting a blank object: p(xt+1 = colored | xt = blank, ut+1 = paint) = 0.9

xt+1 is the state of the object after executing a painting action, ut+1 is the control command, and xt is the state of the object before performing the action.

I know the probability that the sensor indicates that the object is colored although it is blank is p(z = colored | x = blank) = 0.2

I also know that that the Probability that the sensor correctly detects a colored object is p(z = colored | x = colored) = 0.7.

I do not know the current state of the object but after the robot performed a painting action the sensor of the robot indicates that the object is colored. How do I compute the probability that the object is still blank after the robot has performed an action to paint it?

Answer 1

P(xt+1 is blank | z says colored) = P(z says colored|xt+1 is blank)P(xt+1 is blank) / (P(z says colored|xt+1 is blank)P(xt+1 is blank) + P(z says colored|xt+1 is colored)P(xt+1 is colored) = .2*.1/(.2*.1+ .7 * .9) = .02/(.02+ .63) = 2/65