1 problem found
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: \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.