Q.4.46

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?

Step-by-Step Solution

Verified

Answer

$P ($ correct $) \approx 0.87$

$P ($ convinced $) \approx 0.52$

1Step 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

2Step 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 $) = P (X \geq 9, Y = 1) + P (X < 9, Y = 0)$

$= P (X \geq 9 ∣ Y = 1) P (Y = 1) + P (X < 9 ∣ Y = 0) P (Y = 0)$

$= P (X \geq 9 ∣ Y = 1) P (Y = 1) + (1 - P (X \geq 9 ∣ Y = 0)) P (Y = 0)$

3Step 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