<s>[INST] $\frac{2}{3}$ of the people on Weird Island tell the truth all the time and the rest lie all the time 
$\frac{2}{3}$ of the people on Weird Island tell the truth all the
  time and the rest lie all the time. You are sitting in a room with no
  windows and two people come in from outside.
Person 1 says: "It is raining outside"
Person 2 says: "Person 1 is telling the truth"
What is the probability that it is raining outside?

My progress:

 [/INST] According to Bayes theorem. 
$P(E1 = truth) = \frac{1}{2}$
$P(E2 = lie) = 1- \frac{1}{2}$
$P(E2 = lie) = \frac{1}{2}$
$P\left(\frac{A}{E1}\right) = $people speaks truth $=  \frac{2}{3}$
$P\left(\frac{A}{E2}\right) = $people speaks lie $=  1- \frac{2}{3}$
$P\left(\frac{A}{E2}\right) = $people speaks lie $=  \frac{1}{3}$
$Probability = \frac{\frac{A}{E1} * E1}{\frac{A}{E1} * E1 + \frac{A}{E2} * E2}$
$$= \frac{\frac{2}{3} * \frac{1}{2}}{\frac{2}{3} * \frac{1}{2} + \frac{1}{3} * \frac{1}{2}}$$
$= \frac{1}{3} * \frac{2}{1}$
$= \frac{2}{3}$
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