Question

"Professor James Duane from Regent University, USA advocates for constant execution of the right to remain silent and immediately calling the lawyer every time a person is questioned by the police on any topic. He has a popular YouTube lecture "Don’t talk to the Police" and the book "You Have the Right to Remain Innocent". Let’s try to probabilistically evaluate his claims using the following model.

We have individuals with no history of convictions (person NC) and with a history of convictions (person C). Suppose:

• 1) If any person remains silent, there is a 0.001 risk of his case ending up in the court.
• 2) If a person NC talks to the police, there is a 0.002 risk of the police using his words against him and his case ending up in the court.
• 3) If a person C talks to the police, there is a 0.005 risk of the police using his words against him and his case ending up in the court.
• 4) The probability of not being convicted in the court are 0.5 for person NC and 0.1 for person C.
• 5) The probability of not being convicted drops 4 times if a person did not talk to the police during the investigation, as the jury thinks he has something to hide.
• 6) If a person is convicted and talked to the police during the investigation, his sentence is reduced by 0.75 factor as he is considered cooperative.

The police arrested a person suspecting he is involved in a certain crime. The penalty for the crime is 5 years in prison."

Questions:

• a) A person has no history of convictions. What are the expected mean sentence durations he will have to spend in prison if he talks to the police or remains silent?
• b) A person has a history of convictions. What are the expected mean sentence durations he will have to spend in prison if he talks to the police or remains silent?

Answer 1

Part A:

Case: The person has no conviction history(NC) and talks to the police.

Probability of being NC and ended up in court after talking is 0.002.

After getting into court, probability of getting convicted for NC is 0.5.

Then probability of getting convicted if talked to police is 0.002*0.5 = 0.001

The punishment for the crime is 5 years and the duration will decrease by 0.75 factor if the person talks to the police.

So the punishment will be of 5-(5*0.75) = 1.25 years if he talks to the police.

Case: The person has no conviction history(NC) and remains silent.

Probability of ended up in court after remaining silent is 0.001.

After getting into court, probability of getting convicted for NC is 0.5.

The probability of not being convicted drops 4 times if a person did not talk to the police during the investigation. Thus probability of being convicted increases by 4 times if a person remains silent.

Then probability of getting convicted if remains silent is 0.001*0.5*4 = 0.002

The punishment for the crime is 5 years if he remains silent.

Thus the mean sentence duration for a person who has no history of convictions is

(0.001*1.25) + (0.002*5) = 0.01125 years.

Part B:

Case: The person has conviction history(C) and talks to the police.

Probability of being C and ended up in court after talking is 0.005.

After getting into court, probability of getting convicted for C is 0.9.

Then probability of getting convicted if talked to police is 0.005*0.9 = 0.0045

The punishment for the crime is 5 years and the duration will decrease by 0.75 factor if the person talks to the police.

So the punishment will be of 5-(5*0.75) = 1.25 years if he talks to the police.

Case: The person has conviction history(C) and remains silent.

Probability of ended up in court after remaining silent is 0.001.

After getting into court, probability of getting convicted for NC is 0.9.

The probability of not being convicted drops 4 times if a person did not talk to the police during the investigation. Thus probability of being convicted increases by 4 times if a person remains silent.

Then probability of getting convicted if remains silent is 0.001*0.9*4 = 0.0036

The punishment for the crime is 5 years if he remains silent.

Thus the mean sentence duration for a person who has a history of convictions is

(0.0045*1.25) + (0.0036*5) = 0.0236 years.