# Prosecutor’s Fallacy

Prosecutor’s fallacy is a very famous but neglected application of Baye’s rule.

**Prosecutor’s fallacy** is a fallacy in statistical reasoning. This very famous problem uncovers a loop hole in our logical way of thinking. It is confusion between conditional probabilities – probability of A given B and the probability of B given A.

So let’s start and understand what this Prosecutor’s fallacy is all about!

A person commits a crime in a city with a population of say 500000.DNA information is discovered at the crime scene. This information leads to say 10 suspects and one of them is brought to trial. So now we have a defendant in court! **Is He/She innocent?**

To solve this case, we need the following events:

1. **I** : The event that the defendant is innocent.

2. **Ic** : The event that the defendant is guilty.

3. **Ev** : The event that the defendant matches the information collected at the crime scene – Evidence.

The conditional probabilities that corresponds to these events are as follows:

1. **P(Ev|I)** : probability that an innocent person matches the evidence.

2. **P(I|Ev)**: probability that a person who matches the description is innocent.

**The prosecutor makes the following argument :**

A random person has a 1 in 100000 chance of matching the damning evidence. Therefore, if a person has the damning evidence then the person must be guilty.

**In other words:**

An innocent person has a 1 in 100000 chance of matching the damning evidence Ev. Therefore, if the defendant has the damning evidence, there is a 1 in 100000 chance that the defendant is innocent. Which means that the defendant must be guilty.

By making this argument he has committed **Prosecutor’s fallacy**.

Mathematically,

**P(Ev|I) : 1/100000
P(I|Ev) = P(Ev|I) = 1/100000. **

With this probability anyone can state that the person is guilty and must be punished.

**But WAIT!**This probability is not correct. The prosecutor has changed what is uncertain and the condition around. These two probabilities are usually different.

**So what do we do now! How to calculate the correct value of P(I|Ev)?**

The solution is to use **Baye’s rule** to calculate the actual value of **P(I|Ev)**.

**P(I|Ev) = P(Ev|I) * P(I)/P(Ev)**

- What is the probability of P(I|Ev)?
**Illustration**–- Assumptions:
- Guilty person is among the 500000 adults living in the area.
- The Guilty person also matches the Evidence (Ev).

- Probability that a person is innocent:

**P(I)**: 499,999 / 500000 = 0.999998 - Probability that a person is not innocent:

**P(Ic)**: 1 / 500000 = 0.000002 - Probability that the guilty person matches the damning evidence.

**P(Ev|Ic)**: 1 i.e. Guilty person matching evidence is 1 i.e. a 100% - Use Baye’s rule-
**P(Ev) = P(Ev|I)*P(I) + P(Ev|Ic)*P(Ic)**- =0.00001*0.999998 + 1*0.000002
- =0.000012

**P(I|Ev) = P(Ev|I) * P(I)/P(Ev)**- =0.00001 * 0.999998/0.00012

**P(Ic|Ev) = 1 – P(I|Ev)**= 0.16667

- Therefore, there is a
**1/6**chance that a person matching the damning evidence (Ev) is guilty

and a**5/6**chance that a person matching the damning evidence is innocent. **Hence, there is a high chance that the person despite matching the damning evidence is innocent.**

- Assumptions:
- In how many ways the ball will come back to the first boy after N turns
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Source: An Intuitive Introduction to Probability

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