Respuesta :
Solution
- Let M be the event that the machine detects someone is lying
- Let L be the event that the person is actually lying.
- The question tells us that the lie detector will detect a lie 5% of the time if the person is telling the truth.
- This means that the probability that the machine detects a lie given that the person is telling the truth is 5%. This is a conditional probability
- Mathematically, we have:
[tex]\begin{gathered} P(M|L^c)=5\text{ \%}=\frac{5}{100}=0.05 \\ \text{where,} \\ L^c\text{ is the complement of L, that is, the event that the person is telling the truth} \end{gathered}[/tex]- Next, we are told that the machine detects a lie 98% of the time when the person is actually lying.
- This implies that the probability that the machine detects a lie given that the person is lying is 98%.
- Mathematically, we have:
[tex]P(M|L)=98\text{ \%}=\frac{98}{100}=0.98[/tex]- The question asks us for the probability that the detector is completely wrong and gives a positive reading for an innocent suspect and negative reading for the guilty.
- We can also rework this question in familiar conditional probability language as follows:
- We are to find the probability that the lie detector detects a lie given that the innocent person is telling the truth AND the probability that the detector detects nothing for the guilty person given that he lied.
- We should break down this sentence into two aspects and then bring them back together with the AND probability formula. The AND probability formula is given below:
[tex]\begin{gathered} \text{Given events A and B, we have that the AND probability for both events is given as:} \\ P(A\text{ AND }B)=P(A)\times P(B)_{} \end{gathered}[/tex]- Now, let us break down the sentence into two.
Sentence 1:
"We are to find the probability that the lie detector detects a lie given that the innocent person is telling the truth"
This statement corresponds to the probability we had earlier:
[tex]P(M|L^c)=0.05[/tex]Sentence 2:
"...the probability that the detector detects nothing for the guilty person given that he lied."
- This is just the complement of the other probability we had earlier:
[tex]\begin{gathered} \text{ Earlier, we had} \\ P(M|L)=0.98 \\ \text{Now, we have} \\ P(M^c|L) \\ \\ \text{But we know that complements add up to 1} \\ \\ P(M|L)+P(M^c|L)=1 \\ 0.98+P(M^c|L)=1 \\ \text{Subtract 0.98 from both sides} \\ \\ P(M^c|L)=1-0.98 \\ P(M^c|L)=0.02 \end{gathered}[/tex]- Thus, we can now apply the AND probability formula to combine our two results as follows:
[tex]\begin{gathered} \text{if A}=M|L^c\text{ and }B=(M^c|L) \\ \text{Then,} \\ P(A\text{ AND B)}=0.02\times0.05=0.001 \end{gathered}[/tex]Final Answer
The probability that the detector gives a positive reading for the innocent suspect and a negative reading for the guilty suspect, is 0.001
