- #1
Ackbach
Gold Member
MHB
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Problem: A lie detector will show a positive reading (indicate a lie) $10\%$ of the time when a person is telling the truth and $95\%$ of the time when the person is lying. Suppose two people are suspects in a one-person crime and (for certain) one is guilty.
What is the probability that the detector shows a positive reading for both suspects?
Answer: Let $L$ be the event that a suspect is lying, and let $IL$ be the event that a lie detector indicates a lie. We are given the following probabilities:
\begin{align*}
P(IL|L)&=0.95 \\
P(IL|\overline{L})&=0.1.
\end{align*}
Moreover, we may assume that the innocent suspect is telling the truth, and the guilty suspect is lying. We do not know which suspect is guilty and which is innocent, presumably. We may infer that
\begin{align*}
P(\overline{IL}|L)&=0.05 \\
P(\overline{IL}|\overline{L})&=0.9.
\end{align*}
Here's my question: since we don't know which suspect is innocent and which is guilty, should we multiply $P(IL|L) \cdot P(IL| \overline{L})$ by two, since there are two ways to assign the guilty and innocent suspects? Or am I overthinking it here? I'm probably missing something obvious.
Thanks for your time!
What is the probability that the detector shows a positive reading for both suspects?
Answer: Let $L$ be the event that a suspect is lying, and let $IL$ be the event that a lie detector indicates a lie. We are given the following probabilities:
\begin{align*}
P(IL|L)&=0.95 \\
P(IL|\overline{L})&=0.1.
\end{align*}
Moreover, we may assume that the innocent suspect is telling the truth, and the guilty suspect is lying. We do not know which suspect is guilty and which is innocent, presumably. We may infer that
\begin{align*}
P(\overline{IL}|L)&=0.05 \\
P(\overline{IL}|\overline{L})&=0.9.
\end{align*}
Here's my question: since we don't know which suspect is innocent and which is guilty, should we multiply $P(IL|L) \cdot P(IL| \overline{L})$ by two, since there are two ways to assign the guilty and innocent suspects? Or am I overthinking it here? I'm probably missing something obvious.
Thanks for your time!