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#2: Post edited
by
Wolgwang
·
2021-09-02T14:12:22Z (about 3 years ago)
Inserted text and MathJax from the images for better look and search efficiency
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.
>- 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 $0 < w < \frac12 < c < 1$.
- >(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.
- >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 $n \rightarrow \infty$
- [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
DNB
·
2021-07-28T02:15:02Z (over 3 years ago)
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. > Blitzstein, *Introduction to Probability* (2019 2 ed), Example 2.4.5, p 62.