Question

In Example $3f$, suppose that the new evidence is subject to different possible interpretations and in fact shows only that it is $90$ percent likely that the criminal possesses the characteristic in question. In this case, how likely would it be that the suspect is guilty (assuming, as before, that he has the characteristic)?

Short answer

The suspect feel guilty by $87\%$.

Step-by-step solution

Given information

The new evidence is subject to different possible interpretations and in fact shows only that it is$90$ percent likely that the criminal possesses the characteristic in question

Solution

The solution will be,

$G=$event that the suspect is guilty

$C=$event that he possesses the characteristic of the criminal

Then,

$P(G\mid C)=\frac{P\left(GC\right)}{P\left(C\right)}$

$=\frac{\mathrm{P}(\mathrm{C}\mid \mathrm{G})P\left(G\right)}{\mathrm{P}(\mathrm{C}\mid \mathrm{G})P\left(C\mid G^C\right)P\left(G^C\right)}$

$=\frac{\left(0.9\right)\left(0.6\right)}{\left(0.9\right)\left(0.6\right)+\left(0.2\right)\left(0.4\right)}$

$=\frac{0.54}{0.62}$

$=\frac{27}{31}$

$=87\%$

Final answer

In this case that the suspect feel guilty by $87\%$.