Probabilistic Robotics Bayes' Rule I am reading Probabilistic Robotics and I don't know how to solve the first exercise problem at the end of the chapter. There are no solutions to this text. 
Found a copy of the problem online: http://staff.science.uva.nl/~arnoud/education/ProbabilisticRobotics/Assignment2.8.1.pdf
It involves Bayes' Rule which I don't fully understand. Would like some help! 
My efforts:
I figure I have to find P[sensor faulty | sensor output under 1m] 
The probability of the sensor being faulty given that the sensor output is under 1m. 
The probability of the sensor output getting under 1m is 1 * 0.01. 
P(A) = 0.01
 A: It's “[Bayes]“1.
Bayes' rule simply says that: $$p(A|B)=\dfrac{p(B|A)p(A)}{p(B)}.$$
It follows directly by the simmetry of the theorem: $$p(A|B)p(B)=p(A\cap B).$$
A useful way to apply Bayes' rule involves another theorem, which I'm used to call "Alternatives' theorem" (don't know if it's a universal name). It says that, if $E_1,...,E_n$ are a partition of the sample space, then: $$p(A)=\sum _i p(A|E_i)p(E_i).$$
Plugging this into Bayes' rule gives you what is generally called Bayes theorem: $$p(E_i|B)=\dfrac{p(B|E_i)p(E_i)}{\sum _i p(B|E_i)p(E_i)}.$$
In particular, if you take as partition the couple $A,A^c$, (where $A^c$ is "not $A$"), you get:$$p(A|B)=\dfrac{p(B|A)p(A)}{p(B|A)p(A)+p(B|A^c)p(A^c)}$$
A: Expanding on pppqqq's answer, Assuming each sensor reading is independently and identically distributed, you can apply Bayes' theorem for multiple observations:
$\displaystyle P(H_k| E) =  \frac{\prod_j P(E_j|H_k) }{\sum_i P(H_i)\prod_j P(E_j|H_i) }P(H_k)$
Where the $H_k$ is a hypothesis (i.e., faulty or working sensor) of interest and $E_j$ is an individual piece of evidence (i.e., single sensor reading).
For your specific problem that you linked, the probability that a sensor is faulty given $n$ sensor readings below one can be written as:
$\displaystyle P(F| X) =  \frac{P(F)}{P(F) P(X < 1|F)^n + P(W)P(X < 1|W)^n }$
From the problem description, the probability that the sensor is faulty is one in a hundred, $P(F) = 0.01$, thus the probability that it is working is the complement, $P(W) = 0.99$. The probability that a sensor reading is below one given the sensor is faulty is one, $P(X<1|F) = 1$, and when the sensor is working is one in three, $P(X<1|W) = 1/3$.
Thus, the (rounded) probabilities that the sensor is faulty given $N = 1 \ldots 10$ sensor readings below one are listed in the table below.

