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# Falsely accused correctly

## Question

Verdict Is Guilty Verdict is not Guilty

Committed Crime 80% 20% False Negative

Innocent of Crime 10% False Positive 90%

The above chart shows the rate that individuals are correctly found guilty or innocent of crimes in a particular jurisdiction.

If 1% of the population of residents in this county were tried in court, what are the chances that someone was correctly charged with crime?

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8/107 using bayes theorem

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It's the product of the probability that they were actually guilty (not given) times the probability that a guilty person is convicted - which in this case is 8/9.

I think we need an estimate of fraction of people that actually are guilty. Because if that fraction is 0, then there is 0 chance of correctly charging anyone. And if that fraction is 1, there is 100% chance of correctly charging someone.
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It's the product of the probability that they were actually guilty (not given) times the probability that a guilty person is convicted - which in this case is 8/9.

I think we need an estimate of fraction of people that actually are guilty. Because if that fraction is 0, then there is 0 chance of correctly charging anyone. And if that fraction is 1, there is 100% chance of correctly charging someone.

on the left of the chart (committed the crime or innocent of the crime) tells you whether they are actually guilty or not.

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It's the product of the probability that they were actually guilty (not given) times the probability that a guilty person is convicted - which in this case is 8/9.

I think we need an estimate of fraction of people that actually are guilty. Because if that fraction is 0, then there is 0 chance of correctly charging anyone. And if that fraction is 1, there is 100% chance of correctly charging someone.

on the left of the chart (committed the crime or innocent of the crime) tells you whether they are actually guilty or not.

I think the chart says how effective the court is.

I think we need info about the people.

But maybe this puzzle will surprise me.

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It's the product of the probability that they were actually guilty (not given) times the probability that a guilty person is convicted - which in this case is 8/9.

I think we need an estimate of fraction of people that actually are guilty. Because if that fraction is 0, then there is 0 chance of correctly charging anyone. And if that fraction is 1, there is 100% chance of correctly charging someone.

on the left of the chart (committed the crime or innocent of the crime) tells you whether they are actually guilty or not.

I think the chart says how effective the court is.

I think we need info about the people.

But maybe this puzzle will surprise me.

you do not need population numbers, just assume the town is sufficiently large.

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I'll wait to be surprised. Maybe i'm misreading the puzzle.

I don't think you need population estimates; you need guilty estimates.

You ask the probability someone is correctly charged.

If no one in the population has in fact committed a crime, that probability is 0.

If everyone in the population has in fact committed a crime, that probability is 1.

If the OP says on average 2% of the population has committed a crime, then what is ... Etc., the question has an answer.

But again maybe I don't understand the question at all.

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I see the error Bonanova was talking about I meant that 1% of the population actually committed a crime (sorry, i am still naive in thinking only criminals are tried in court )

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dm92 has it.

Suppose the town's population is 1000, then 10 of them committed a crime (1%)
Now suppose they are all tried for the crime.

Convicted Acquitted
10 did the crime 8 2
990 are good guys 99 891
Total 107 893

If the whole town were tried, there would be 107 convictions.
Of those, only 8 would be rightly convicted.

So the probability that a person who was found guilty actually did the deed
is a paltry 8/107 = 0.0747 (less than an 8% probability of a correct conviction.
)

This type of analysis also applies to things like drug testing in a

population where the vast majority of those tested are clean.

The positive results will be dominated by false positives.

The difficulty can be alleviated simply by repeat testing those who tested positive once.

Now the population has a higher fraction of actual offenders, and the results are more reliable.

The false positives will be greatly reduced relative to the true positives.

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