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P (X = n | G), P (X = n | G^{C})

by the Law of Total Probability, with extra conditioning?

0 answers · posted 3y ago by DNB‭ · edited 3y ago by Wolgwang‭

Question probability

#2: Post edited by $user avatar$ Wolgwang‭ · 2021-09-02T14:12:22Z (over 3 years ago)
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How do you calculate $P(X = n|G), P(X = n|G^C)$ by the Law of Total Probability, with extra conditioning?

1. Please see $P (U | G)$ and $P (U | G^{C})$ below, beside my red line. Can you please expound these calculations?
2. How was $s$ computed in both equations?
>### Example 2.4.5 (Unanimous agreement).
>The article "Why too much evidence can
be a bad thing" by Lisa Zyga [30] says:
>>Under ancient Jewish law, if a suspect on trial was unanimously found guilty
by all judges, then the suspect was acquitted. This reasoning sounds counterintu-
itive, but the legislators of the time had noticed that unanimous agreement often
indicates the presence of systemic error in the judicial process.
>There are n judges deciding a case. The suspect has prior probability p of being
guilty. Each judge votes whether to convict or acquit the suspect. With probability s,
a systemic error occurs (e.g., the defense is incompetent). If a systemic error occurs,
then the judges unanimously vote to convict (i.e., all n judges vote to convict).
>Whether a systemic error occurs is independent of whether the suspect is guilty.
Given that a systemic error doesn't occur and that the suspect is guilty, each judge
has probability c of voting to convict, independently. Given that a systemic error
doesn't occur and that the suspect is not guilty, each judge has probability w of
voting to convict, independently. Suppose that
> $0 < p < 1; 0 < s < 1;$ and $0 < w < \frac{1}{2} < c < 1$ .
>(a) For this part only, suppose that exactly k out of n judges vote to convict, where
k < n. Given this information, find the probability that the suspect is guilty.
>(b) Now suppose that all n judges vote to convict. Given this information, find the
probability that the suspect is guilty.
>(c) Is the answer to (b), viewed as a function of n, an increasing function? Give a
short, intuitive explanation in words.
~~>![Image alt text](https://math.codidact.com/uploads/eEUGmJtT4WZjdTfJRWpU4bxY)~~
Blitzstein, *Introduction to Probability* (2019 2 ed), Example 2.4.5, p 62.

1. Please see $P (U | G)$ and $P (U | G^{C})$ below, beside my red line. Can you please expound these calculations?
2. How was $s$ computed in both equations?
>### Example 2.4.5 (Unanimous agreement).
>The article "Why too much evidence can
be a bad thing" by Lisa Zyga [30] says:
>>Under ancient Jewish law, if a suspect on trial was unanimously found guilty
by all judges, then the suspect was acquitted. This reasoning sounds counterintu-
itive, but the legislators of the time had noticed that unanimous agreement often
indicates the presence of systemic error in the judicial process.
>There are n judges deciding a case. The suspect has prior probability p of being
guilty. Each judge votes whether to convict or acquit the suspect. With probability s,
a systemic error occurs (e.g., the defense is incompetent). If a systemic error occurs,
then the judges unanimously vote to convict (i.e., all n judges vote to convict).
>Whether a systemic error occurs is independent of whether the suspect is guilty.
Given that a systemic error doesn't occur and that the suspect is guilty, each judge
has probability c of voting to convict, independently. Given that a systemic error
doesn't occur and that the suspect is not guilty, each judge has probability w of
voting to convict, independently. Suppose that
> $0 < p < 1; 0 < s < 1;$ and $0 < w < \frac{1}{2} < c < 1$ .
>(a) For this part only, suppose that exactly k out of n judges vote to convict, where
k < n. Given this information, find the probability that the suspect is guilty.
>(b) Now suppose that all n judges vote to convict. Given this information, find the
probability that the suspect is guilty.
>(c) Is the answer to (b), viewed as a function of n, an increasing function? Give a
short, intuitive explanation in words.
>Solution:
(a) Since $k < n$ , a systemic error didn't occur. We will implicitly condition on this in this part. Let $G$ be the event that the suspect is guilty and $X$ be the number of judges who vote to convict. Using Bayes' rule, LOTP, and the Binomial PMF,
$$
P(G \mid X=k)=\frac{P(X=k \mid G) P(G)}{P(X=k)}=\frac{p c^{k}(1-c)^{n-k}}{p c^{k}(1-c)^{n-k}+(1-p) w^{k}(1-w)^{n-k}}
$$
(b) Let $U$ be the event $X = n$ and $B$ be the event that a systemic error occurs. Then
$$
P(G \mid U)=\frac{P(U \mid G) P(G)}{P(U)}=\frac{p P(U \mid G)}{p P(U \mid G)+(1-p) P\left(U \mid G^{c}\right)}
$$
By LOTP with extra conditioning,
<p>
$$
\color{red}{\begin{aligned}
P(U \mid G) &=P(U \mid G, B) P(B \mid G)+P\left(U \mid G, B^{c}\right) P\left(B^{c} \mid G\right)=s+(1-s) c^{n} \
P\left(U \mid G^{c}\right) &=P\left(U \mid G^{c}, B\right) P\left(B \mid G^{c}\right)+P\left(U \mid G^{c}, B^{c}\right) P\left(B^{c} \mid G^{c}\right)=s+(1-s) w^{n}
\end{aligned}}
$$
</p>
> Thus,
$$
P(G \mid U)=\frac{p\left(s+(1-s) c^{n}\right)}{p\left(s+(1-s) c^{n}\right)+(1-p)\left(s+(1-s) w^{n}\right)}
$$
(c) No, since a large value of $n$ yields a high chance of systemic error, and if a systemic error occurs then the judges' votes are uninformative about whether the suspect is guilty. The answer to (b) reverts to the prior probability $p$ as $n \to \infty$
[Original image](https://math.codidact.com/uploads/eEUGmJtT4WZjdTfJRWpU4bxY)
Blitzstein, *Introduction to Probability* (2019 2 ed), Example 2.4.5, p 62.

#1: Initial revision by $user avatar$ DNB‭ · 2021-07-28T02:15:02Z (over 3 years ago)

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How do you calculate $P(X = n|G), P(X = n|G^C)$ by the Law of Total Probability, with extra conditioning?

1. Please see  $P(U|G)$ and $P(U|G^C)$ below, beside my red line. Can you please expound these calculations? 

2. How was $s$ computed in both equations? 



>### Example 2.4.5 (Unanimous agreement). 

>The article "Why too much evidence can
be a bad thing" by Lisa Zyga [30] says:

>>Under ancient Jewish law, if a suspect on trial was unanimously found guilty
by all judges, then the suspect was acquitted. This reasoning sounds counterintu-
itive, but the legislators of the time had noticed that unanimous agreement often
indicates the presence of systemic error in the judicial process.

>There are n judges deciding a case. The suspect has prior probability p of being
guilty. Each judge votes whether to convict or acquit the suspect. With probability s,
a systemic error occurs (e.g., the defense is incompetent). If a systemic error occurs,
then the judges unanimously vote to convict (i.e., all n judges vote to convict).

>Whether a systemic error occurs is independent of whether the suspect is guilty.
Given that a systemic error doesn't occur and that the suspect is guilty, each judge
has probability c of voting to convict, independently. Given that a systemic error
doesn't occur and that the suspect is not guilty, each judge has probability w of
voting to convict, independently. Suppose that

>$0 < p < 1; 0 < s < 1;$ and $0 < w < \frac12 < c < 1$.

>(a) For this part only, suppose that exactly k out of n judges vote to convict, where
k < n. Given this information, find the probability that the suspect is guilty.

>(b) Now suppose that all n judges vote to convict. Given this information, find the
probability that the suspect is guilty.

>(c) Is the answer to (b), viewed as a function of n, an increasing function? Give a
short, intuitive explanation in words.

>![Image alt text](https://math.codidact.com/uploads/eEUGmJtT4WZjdTfJRWpU4bxY)

Blitzstein, *Introduction to Probability* (2019 2 ed), Example 2.4.5, p 62.

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