Cambridge Question - The Prosecutor's Fallacy.

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EXERCISE 4B Question 13 - Statistics 1 (Steve Dobbs & Jane Miller)

"The Prosecutor's Fallacy. An accused prisoner is on trial. The defense lawyer asserts that, in the absence of further evidence, the probability that the prisoner is guilty is **1 in a million**. The prosecuting lawyer produces a further piece of evidence and asserts that, if the prisoner were guilty, the probability that this evidence would be obtained is **999 in 1000**, and if he were not guilty it would be only **1 in 1000**; in other words, $\displaystyle P(evidence|guilty)= 0.999$, and $\displaystyle P(evidence|not guilty)= 0.001$. Assuming that the court admits the legality of the evidence, and that both lawyers' figures are correct, what is the probability that the prisoner is guilty?"

Book Answer: **0.000998**

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This is one of the last few challenging and out of syllabus question in my textbooks. I am confused about ways to tackle it, whether this need Bayes' theorem, some linear solution or simply by Tree diagram.

I have been desperately listening Alexandra Stan (Million) all day, atleast I won't forget the question and you can do same, trying to find a solution to that problem and hoping someone to clearly show a solution. Thank you.

Alexandra Stan feat Carlprit - Million - Official Video - YouTube

Re: Cambridge Question - The Prosecutor's Fallacy.

Hey zikcau25.

Hint: P(Evidence|guilty) + P(Evidence|not guilty) = P(Evidence). P(Guilty) = 1/1000000. P(Evidence and Guilty) = P(Evidence|Guilty)*P(Guilty).

What is P(Guilty|Evidence) equal to? (Remember also that P(A|B) = (P(B|A)*P(A))/P(B))

Re: Cambridge Question - The Prosecutor's Fallacy.

Master Chiro, thank you so much for guiding me to the correct answer. All credits goes to you but it contradicts two of your proposals.

Firstly,

$\displaystyle P(Evidence|Guilty) + P(Evidence|not Guilty) \neq P(Evidence)$

(Checked using numerical values*, IF P(A)=0.75, P(B|A)=0.8, P(B|A')=0.6, P(B)=0.75, P(A|B)=0.8 then ***P(B|A) + P(B|A')**** ≠ ***P(B)* as 0.8 + 0.6 ≠ 0.75)

Secondly,

The probability of the prisoner is guilty is interpreted by **P(Guilty|Evidence)** instead of P(Evidence and Guilty)

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Here my solution,

(expressed in terms of **G** for **Guilty**, **G'** for **not Guilty**, **E** for **Evidence**, **E'** for **no Evidence**).

**Bayes' Theorem** says:

**1.** Simple form:

$\displaystyle P(E|G)=\frac{P(E)\cdot P(G|E)}{{\color{DarkBlue} P(G)}}$

$\displaystyle P(G|E)=\frac{P(G)\cdot P(E|G)}{{\color{DarkBlue} P(E)}}$

**2.** Extended form:

$\displaystyle P(E|G)=\frac{P(E)\cdot P(G|E)}{{\color{DarkBlue} P(E)\cdot P(G|E)+P(E')\cdot P(G|E')}}$

$\displaystyle P(G|E)=\frac{P(G)\cdot P(E|G)}{{\color{DarkBlue} P(G)\cdot P(E|G)+P(G')\cdot P(E|G')}}$

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Now, referring back to our initial question,

Given that:

$\displaystyle P(G)=\frac{1}{10^{6}}$, so $\displaystyle P(G')=\frac{999999}{10^{6}}$

$\displaystyle P(E|G)=0.999$

$\displaystyle P(E|G')=0.001$

Evidently, the second equation of the extended form $\displaystyle P(G|E)=\frac{P(G)\cdot P(E|G)}{{\color{DarkBlue} P(G)\cdot P(E|G)+P(G')\cdot P(E|G')}}$ requires just all the values that we have got.

Substituting the values,

$\displaystyle P(G|E)=\frac{\frac{1}{10^{6}}\times 0.999}{\frac{1}{10^{6}}\times 0.999+\frac{999999}{10^{6}}\times 0.001}$**= 0.000 998** (3 sig.)

Re: Cambridge Question - The Prosecutor's Fallacy.

I really hate using long, complicated, formulas like that! Here is how I would do it:

Imagine a population of 1,000,000,000 people (chosen to avoid fractions). Since "in the absence of further evidence, the probability that the prisoner is guilty is 1 in a million", 1000 of those people are guilty. Of those 1000 people, 999 would have this particular "evidence" against them. Of the 999,991,000 who are not guilty, 1 in a thousand, 999,991 would have this evidence. That is, of the total 999+ 999,991= 1000990 people against whom we could find this evidence, 999 are guilty. The probability that a person, against whom we have that evidence, is guilty is [tex]\frac{999}{1000990}= 0.000998[/quote] as you say.