Decision Making Under Uncertainty

A slither of light from a nearby business sign invaded the room through slightly parted drapes.

The turbulent stomach, that drove me to bed, returned. Now my stomach began to squirm and knot.

Slowly my mind focused then it locked on, "I'm in a hotel room, sequestered with 11 others. We are jurors in a felony trial of an infamous criminal."

The Stakes: Life In Prison.

My stomach tightens even more.

The drama of last evening flashed on my mind's screen.

Another juror and I voted "not guilty" during the initial poll to commence deliberation. Arguments and emotions were fierce, but we held our ground.

Something was not right in the majority's argument. The pieces did not fall in place. So I announced my divergence from the majority opinion.

However, my voting partner's eyes signaled he could not withstand another onslaught like last nights'. I realized I am in the eye of the hurricane. In a few hours the force and power of the majority will strike again with even greater vengeance. What am I to do?

The Trial Data
James Korovopt, a member of a local gang that terrorizes and extorts defenseless seniors, is alleged to have illegally entered the home of a cripple elderly at gun point, and to have shot the elderly when he refused to surrender his savings account information.

For the first time, the state has the best circumstantial case against this wiry outlaw. The District Attorney is cautious, but he flashes a confidence announcing prison is imminent.

The defense is also confident.

It presented two eye-witness testimonies placing Korovopt elsewhere during the home invasion. The two witnesses do not know one another, neither have they had contact with Korovopt, nor with each other since the trial began.

To the courts satisfaction, the District Attorney established that the eyewitnesses tell the truth about 10% of the time. However, the defense argued their testimonies should has value because they gave identical relevant details.

If the jury buys the witnesses' account, Korovopt will go free.

My Background -- My Attack
Having received an education that valued mathematics and critical thinking, I decided to defend my position during the next deliberation.

My education stressed logical connections are the most important elements in a mathematical statement. Further, the concepts in the mathematical statement, to which these logical connections apply, are mere placeholders; for other concepts.

With these foundational building blocks, I constructed an urn model of the situation - careful to model the important problem elements while eliminating the distracting ones.

Building The Analysis
For each potential testimony, I placed a colored ball in the urn. Except for color, the balls are indistinguishable; same size, same weight, and same texture.

To the testimony that places the defendant in a place making it impossible to have committed the crime, I associate a white ball.

Then, to represents a witness' testimony, I modeled Fate selecting a ball from the urn representing a witness' testimony.

Since both witnesses gave a testimony freeing the defendant, thus Fate selected the white ball.

I decided to base my decision on the results of the probability that "Fate selected the white ball when these two observers said it is white".

If the probability is close to zero, I will vote guilty. Otherwise, I will vote not guilty.

The Probability
I calculated the probability that the ball is in fact white, when these less than truthful witnesses say it is white. The probability is (n-1)/(n+80).

I noticed for 1000 possible testimonies, the probability is (1000-1)/(1000+80) = 999/1080. This is almost certainty.

I convinced myself that a large number of possible testimonies is open to the witnesses. So I embrace these witnesses' convergence as truth.

The phone rings. I awake.

My carpool colleague frantically reminds me it is my turn to drive and I should have arrived five minutes ago. I wipe the sleep from my face and began to assemble my things for work, thinking, "No more spicy chili and beer before bed. It causes to much thought."

Throughout the day, the problem stirred in my head.

Epilogue
In the article Elements de la theorie des probabilities, Emile Borel (1871-1956) a pioneering 20th century mathematician, believed the best approach to analyzing observed data is to construct urn models, having fixed compositions, such that drawings from the urn identically replicate the table of observed values.

We now know this approach will not be satisfactory for many interesting cases. But it proves to be valuable in many decision-making inquiries; as it is in this case.

One of the goals of education is to aid a child to gain dominion over his world. An education equipping a child with critical thinking skills and facility with mathematics and sciences gives that advantage. Such an education teaches a child the elements of this analysis, thus it should be every parent's desire for his child.

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The Problem
The urn problem in this note appears in Exercises in Probability and Statistics by N. A. Rahman p 4. It was a private communication from H.D. Downton. There it is stated:

An urn contains n balls each identical in size and texture but with different colors. One of the balls is white. Two independent observers, each having a probability of 0.1 of telling the truth, assert that a ball drawn at random from the urn is white. The probability that the ball is in fact white is (n-1)/(n+80).