Question

Suppose that it takes at least $9$votes from a $12$- member jury to convict a defendant. Suppose also that the probability that a juror votes a guilty person innocent is $.2,$ whereas the probability that the juror votes an innocent person guilty is $.1.$ If each juror acts independently and if $65$ percent of the defendants are guilty, find the probability that the jury renders a correct decision. What percentage of defendants is convicted?

Short answer

$P($correct $)\approx 0.87$

$P($convinced $)\approx 0.52$

Step-by-step solution

Step 1:Given information

Suppose that it takes at least $9$votes from a $12$- member jury to convict a defendant. Suppose also that the probability that a juror votes a guilty person innocent is .$2,$whereas the probability that the juror votes an innocent person guilty is .$1.$ If each juror acts independently and if $65$ percent of the defendants are guilty, find the probability that the jury renders a correct decision

Step 2:Explanation

Define random variable $X$that marks the number of jurors that vote that the defendant is guilty. Also, define $Y$that marks whether the defendant is guilty or not. The jury will make the right decision if and only if they convict a guilty person or they do not convict an innocent person. Hence

$P($correct $)=P(X\ge 9,Y=1)+P(X<9,Y=0)$

$=P(X\ge 9\mid Y=1)P(Y=1)+P(X<9\mid Y=0)P(Y=0)$

$=P(X\ge 9\mid Y=1)P(Y=1)+(1-P(X\ge 9\mid Y=0\left)\right)P(Y=0)$

Step 3:Explanation

$P(Y=0)=0.35$Observe that if$Y$is given, then we know that every juror votes guilty independently from every other with probabilities $0.8$if $Y=1$and $0.1$if $Y=0$. Hence, given $Y,X$has binomial distribution. We have that

$P(X\ge 9\mid Y=1)\approx 0.795$

$P(Y=1)=0.65$

so it yields

$P($correct $)\approx 0.87$

The percentage of convinced is

$P($convinced $)=P(X\ge 9\mid Y=1\left)P\right(Y=1)+P(X\ge 9\mid Y=0\left)P\right(Y=0)\approx 0.52$

Step 4:Final answer

$P($correct $)\approx 0.87$

$P($convinced$)$$\approx 0.52$