Problem source

Mr Blond returns to his flat to find it in complete darkness. He knows that this means that one of four assassins Mr 1, Mr 2, Mr 3 or Mr 4 has set a trap for him. His trained instinct tells him that the probability that Mr $i$ has set the trap is $i/10$. His knowledge of their habits tells him that Mr $i$ uses a deadly trained silent anaconda with probability $(i+1)/10$, a bomb with probability $i/10$ and a vicious attack canary with probability $(9-2i)/10$ $[i=1,2,3,4]$.
 
He now listens carefully and, hearing no singing, concludes correctly that no canary is involved. If he switches on the light and the trap is a bomb he has probability $1/2$  of being killed
but if the trap is an anaconda he has probability $2/3$ of survival. If he does not switch on the light and the trap is a bomb he is certain to survive but, if the trap is an anaconda, he has a probability $1/2$ of being killed. His professional pride means that he must enter the flat. 
Advise Mr Blond, giving reasons for your advice.

Solution source

\begin{array}{c|c|c|c} 
& A & B & C \\ \hline
1 & \frac{1}{10} \cdot \frac{2}{10}  & \frac{1}{10} \cdot \frac{1}{10} & \frac{1}{10} \cdot \frac{7}{10} \\
2 & \frac{2}{10} \cdot \frac{3}{10}  &\frac{2}{10} \cdot \frac{2}{10} &\frac{2}{10} \cdot \frac{5}{10}  \\
3 & \frac{3}{10} \cdot \frac{4}{10}  &\frac{3}{10} \cdot \frac{3}{10} &\frac{3}{10} \cdot \frac{3}{10}  \\
4 & \frac{4}{10} \cdot \frac{5}{10}  &\frac{4}{10} \cdot \frac{4}{10} &\frac{4}{10} \cdot \frac{1}{10}  \\ \hline
& \frac{2+6+12+20}{100} & \frac{1 + 4 + 9 + 16}{100} & \frac{7 + 10 + 9 + 4}{100} 
\end{array}

Therefore $\mathbb{P}(A) = \frac{4}{10}, \mathbb{P}(B) = \frac{3}{10}, \mathbb{P}(C) = \frac{3}{10}$, in particular,

\begin{align*}
\mathbb{P}(A | \text{not }C) &= \frac{4}{7} \\
\mathbb{P}(B | \text{not }C) &= \frac{3}{7} \\
\end{align*}

If he switches the light on, his probability of survival is $\frac47 \cdot \frac23 + \frac37 \cdot \frac12 = \frac{25}{42}$, if he doesn't his probability is $\frac12 \cdot \frac47 +\frac37= \frac{5}{7} = \frac{30}{42}$ therefore he shouldn't switch the light on.