The Monty Hall Problem

The Monty Hall problem is one that you may be familiar with already (and you probably do not even know it).

Some of you may remember this scene in a recent movie:

The problem has been debated by statisticians for decades.

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My take on the Monty Hall problem:

You are on a game show and the host asks you to pick from three doors (one door has a car behind it). You pick door #1. The host opens door #3 (which uncovers a goat behind it) and offers you the option to change your choice from door #1 to door #2.

At the beginning you had a 33% of guessing the door that hides a car behind it. After the host opened door #3 (and showed you the goat) only two possibilities remain: door #1 and door #2. That’s a 50% chance that either door has a car behind it. When the host asks if you would like to change your choice to door #2, a new game has officially begun.

Bayes’ theorem, as it applies to the Monty Hall problem is still appicable. Accept now, where the denominator is computed using the law of total probability as the marginal probability as seen here:

is inapplicable.

Why? Because the marginal probability cannot be measured yet in this new game (and is unnecessary because the solution will be presented by the host after affirming choosing door #1 or door #2).

…in other words, switching your choice to door #2 will give you the same odds (50%) as keeping your original choice.

In the first game there were three doors. In this new game there are only two doors.

If you consider this game as a new one, then switching your original choice from door #1 to door #2 (as encouraged by Vos Savant) now seems arbitrary.

Conceptual Vision to Limit Citizen’s Privacy Invasion by the US Government (Part 1)

Popular sites like Facebook, LinkedIn, YouTube, and others are compiling tremendous amounts of personal data on their users.

Imagine the depth of information the United States Government is capable of retrieving on a person. The FBI, CIA, IRS, and other intelligence agencies (all funded by taxpayers) undoubtedly host an even more complex consortium of information.

Such agencies’ rights to compile this data is an issue in itself. But the reason for this blog is to address the due process procedures that need to be re-written (and adhered by the [once-citizens/now-]agents of all our government branches).

What can you tell about this person?

Two questions I have for each US Citizen:

Do you feel our government already knows more than you would like them to about you?

Do you feel it is reasonable to set new laws that require government agencies to follow clearly defined procedures before prying into personal aspects about your life?

 

I will try to pass this point with a more pathos-infused real-world situation:

In a municipal court case resting on the key testimony of Officer JOE, who arrested Citizen TOM for negligent driving, the case has been pushed to trail. A jury of Citizen TOM’s peers will decide his fate – whether or not TOM will receive a charge of negligent driving on his permanent record (and any applicable fines, community service, and/or jail time).

Before giving testinomey in front of TOM’s peers, Officer JOE coordinates an investigation in coordination with his friends in the Prosecuting Attorneys Office. Fortunately for JOE, the reach of his investigation goes much further than the filing cabinets in the PA’s office. This is because the Data Records System used by the PA’s office are shared with many other prominent government agencies.

In this particular case, the PA’s office is able to uncover a small treasure trove of private images (displaying TOM with his friends around their fast cars), two YouTube Videos titled, “How to Drift on Interstate-5 Without Getting Busted” and “My car can reach 190mph, can yours?”, as well as several personal emails between TOM and his friends regarding past racing meetups.

When Officer JOE takes the stand he is questioned about the public service he has provided to his community as an officer, his past awards and recommendations, the time he saved the mayor’s daughter from a kidnapping, and oh, briefly he is asked about his observation of TOM’s negligent driving. Once Office JOE steps down from the stand, the PA presents the ‘uncovered’ incriminating evidence to the jury.

When TOM takes the stand he still in shock about the evidence that has just been presented against him. He tells the jury that the racing videos from YouTube were taken when he was 16, deleted shortly after, and are anecdotal pieces of evidence that are now almost 8 years old. A similar explanation of the emails and photos is given by TOM to the jury as well.

All of the evidence, TOM claims, was ‘permanently deleted’ many years ago while TOM began seeking employment after graduating from college. Lastly, TOM tells the jurors that he has never received any sort of traffic or parking ticket before the one cited by Officer JOE and that he was making the improper lane changes (that constitute his charge of negligent driving) on the day that Officer JOE cited him.

The jury hands down a verdict of ‘guilty’ after only four hours of deliberation.

What happened in this hypothetial court case?

Can you see where the evidence held against TOM was done so unfairly?