A theft is committed by one of two suspects, Aaron and Becca.
Initially, there is equal evidence against both of them. In further
investigation at the crime scene, it is found that the guilty
individual had a blood type found in 10% of the population.
Suspect Aaron does match this blood type, whereas the blood type
of Suspect Becca is unknown.
a) Given this new information, what is the probability that
Aaron is the guilty party?
b) Given this new information, what is the probability that
Becca’s blood type matches that found at the crime scene?
for part one I have arrived at the following solution
P(A) = Aaron is guilty = 0.5
P(B)= $P(A^c)$ = Becca is guilty = 0.5
P(G) = guilty individual has a blood type = 0.1
other information
Aaron matched the blood type found in guilty individuals
part a
using bayes rule
$ P(A/G) = \frac{P(G/A).P(A)}{P(G)}$
applying law of total probability
$ P(A/G) = \frac{P(G/A).P(A)}{P(G/A).P(A) + P(G/A^c).P(A^c)}$
$P(G/A) = 1$ since A's blood matches with those found in guilty population
$P(G/A^c) = P(G/B)= 0.1$ Since the suspect has 10% chanes of having the blood found in guilty.
$ P(A/G) = \frac{1*0.5)}{1*0.5 + 0.1*0.5} = \frac{10}{11} = 0.90$
any hints about the second part which asks for the probability that beccas blood matches that found at the crime scene given that the guilty have 10 percent chances of having that blood . thanks
Best Answer
Calculations are correct, but your logic has gone a bit awry.
No. The guilty individual certainly has the blood type (it is evidence). An innocent person has that type with probability $0.10$.
That is, the probability the criminal's blood type matches Aaron when given guilt is $\mathsf P(M_A\mid A)=1$, but when given innocence is $\mathsf P(M_A\mid A^\complement)=0.1$ (assuming blood type is independent of all other incriminating evidence).
Now find $\mathsf P(A\mid M_A)$ by the usual methods
$$\mathsf P(A\mid M_A) = \dfrac{\mathsf P(A)\mathsf P(M_A\mid A)}{\mathsf P(A)\mathsf P(M_A\mid A)+\mathsf P(A^\complement)\mathsf P(M_A\mid A^\complement)} = \dfrac{1}{1+0.1}=0.\dot{\overline{90}}$$
Then $\mathsf P(B\mid M_A)=0.\dot{\overline{09}}$ and, by total probability, the probability that the blood type will match Becca when given that it matches Aaron is::
$$\begin{align}\mathsf P(M_B\mid M_A) ~=~& \mathsf P(M_B\mid A)\mathsf P(A\mid M_A)+\mathsf P(M_B\mid B)\mathsf P(B\mid M_A) \\[1ex] \vdots~& \\[1ex]~=~& \bbox[white]{\color{white}{\frac 2{11}}}\end{align}$$