Q.4.47 - A First Course in Probability

Question

In some military courts, $9$ judges are appointed. However, both the prosecution and the defense attorneys are entitled to a peremptory challenge of any judge, in which case that judge is removed from the case and is not replaced. A defendant is declared guilty if the majority of judges cast votes of guilty, and he or she is declared innocent otherwise. Suppose that when the defendant is, in fact, guilty, each judge will (independently) vote guilty with probability .$7,$whereas when the defendant is, in fact, innocent, this probability drops to .$3.$ 

(a) What is the probability that a guilty defendant is declared guilty when there are (i) $9$, (ii) $8$, and (iii) $7$judges?

(b) Repeat part (a) for an innocent defendant. 

(c) If the prosecuting attorney does not exercise the right to a peremptory challenge of a judge, and if the defense is limited to at most two such challenges, how many challenges should the defense attorney make if he or she is $60$ percent certain that the client is guilty? 

Step-by-Step Solution

Answer

Defendant has minimal probability that he will be judged if we remove one juror.

Step 1: Given information (part a)

In some military courts, $9$ judges are appointed. However, both the prosecution and the defense attorneys are entitled to a peremptory challenge of any judge, in which case that judge is removed from the case and is not replaced. A defendant is declared guilty if the majority of judges cast votes of guilty, and he or she is declared innocent otherwise. Suppose that when the defendant is, in fact, guilty, each judge will (independently) vote guilty with probability .$7,$ whereas when the defendant is, in fact, innocent, this probability drops to .$3.$ 

Step 2:Explanation(part a)

Define random variable $X$ that marks the number of jurors that vote that the defendant is guilty and suppose that there are $n$ jurors. Also, define$Y$that marks whether the defendant is guilty or not.

Given $Y$, we have that $X$ has binomial distribution with parameters $n$and appropriate probability of judgement ( $0.7$ if $Y=1$ and $0.3$ otherwise). Hence

$n=9\Rightarrow P(X\ge 5\mid Y=1)=\sum _{i=5}^9\left(\begin{array}{l}9\\ i\end{array}\right)0.7^{i}0.3^{9-i}\approx 0.90119134$

$n=8\Rightarrow P(X\ge 5\mid Y=1)=\sum _{i=5}^8\left(\begin{array}{l}8\\ i\end{array}\right)0.7^{i}0.3^{8-i}\approx 0.80589565$

$n=7\Rightarrow P(X\ge 4\mid Y=1)=\sum _{i=4}^7\left(\begin{array}{l}7\\ i\end{array}\right)0.7^{i}0.3^{7-i}\approx 0.873964$

Step 3:Final answer(part a)

$n=9\Rightarrow P(X\ge 5\mid Y=1)=\sum _{i=5}^9\left(\begin{array}{l}9\\ i\end{array}\right)0.7^{i}0.3^{9-i}\approx 0.90119134$

$n=8\Rightarrow P(X\ge 5\mid Y=1)=\sum _{i=5}^8\left(\begin{array}{l}8\\ i\end{array}\right)0.7^{i}0.3^{8-i}\approx 0.80589565$

$n=7\Rightarrow P(X\ge 4\mid Y=1)=\sum _{i=4}^7\left(\begin{array}{l}7\\ i\end{array}\right)0.7^{i}0.3^{7-i}\approx 0.873964$

Step 4:Given information (part b)

In some military courts, $9$judges are appointed. However, both the prosecution and the defense attorneys are entitled to a peremptory challenge of any judge, in which case that judge is removed from the case and is not replaced. A defendant is declared guilty if the majority of judges cast votes of guilty, and he or she is declared innocent otherwise. Suppose that when the defendant is, in fact, guilty, each judge will (independently) vote guilty with probability$.7$, whereas when the defendant is, in fact, innocent, this probability drops to $.3$. 

Step 5:Explanation(part b)

Define random variable $X$ that marks the number of jurors that vote that the defendant is guilty and suppose that there are $n$ jurors. Also, define$Y$that marks whether the defendant is guilty or not.

Similarly as in (a), but here we are given that $Y=0$. We have that

$n=9\Rightarrow P(X\ge 5\mid Y=0)=\sum _{i=5}^9\left(\begin{array}{l}9\\ i\end{array}\right)0.3^{i}0.7^{9-i}\approx 0.09880866$

$n=8\Rightarrow P(X\ge 5\mid Y=0)=\sum _{i=5}^8\left(\begin{array}{l}8\\ i\end{array}\right)0.3^{i}0.7^{8-i}\approx 0.05796765$

$n=7\Rightarrow P(X\ge 4\mid Y=0)=\sum _{i=4}^7\left(\begin{array}{l}7\\ i\end{array}\right)0.3^{i}0.7^{7-i}\approx 0.126036$

Step 6:Final answer(part b)

$n=9\Rightarrow P(X\ge 5\mid Y=0)=\sum _{i=5}^9\left(\begin{array}{l}9\\ i\end{array}\right)0.3^{i}0.7^{9-i}\approx 0.09880866$

$n=8\Rightarrow P(X\ge 5\mid Y=0)=\sum _{i=5}^8\left(\begin{array}{l}8\\ i\end{array}\right)0.3^{i}0.7^{8-i}\approx 0.05796765$

$n=7\Rightarrow P(X\ge 4\mid Y=0)=\sum _{i=4}^7\left(\begin{array}{l}7\\ i\end{array}\right)0.3^{i}0.7^{7-i}\approx 0.126036$

Step 7:Given information(part c)

In some military courts, $9$judges are appointed. However, both the prosecution and the defense attorneys are entitled to a peremptory challenge of any judge, in which case that judge is removed from the case and is not replaced. A defendant is declared guilty if the majority of judges cast votes of guilty, and he or she is declared innocent otherwise. Suppose that when the defendant is, in fact, guilty, each judge will (independently) vote guilty with probability .$7,$whereas when the defendant is, in fact, innocent, this probability drops to $.3.$

If the prosecuting attorney does not exercise the right to a peremptory challenge of a judge, and if the defense is limited to at most two such challenges 

Step 8: Explanation(part c)

Define random variable $X$ that marks the number of jurors that vote that the defendant is guilty and suppose that there are $n$ jurors. Also, define$Y$ that marks whether the defendant is guilty or not.

Step 9:Explanation(part c)

We here have that $P(Y=1)=0.6$. Let's look if we remove none, one or two jurors. If we do not remove any of them, we have that the defendant will be judged with the probability (input the case $n=9$)

$P(X\ge 5\mid Y=0)P(Y=0)+P(X\ge 5\mid Y=1)P(Y=1)=0.58$

If we remove one juror, we have that (input case $n=8$)

$P(X\ge 5\mid Y=0)P(Y=0)+P(X\ge 5\mid Y=1)P(Y=1)=0.51$

If we remove two jurors, we have that (input case $n=7$)

$P(X\ge 4\mid Y=0)P(Y=0)+P(X\ge 4\mid Y=1)P(Y=1)=0.57$

Observe that the defendant has minimal probability that he will be judged if we remove one juror.

Step 10: Final answer

Defendant has minimal probability that he will be judged if we remove one juror.