Wason Selection Task

One of the most intriguing instances where our brain goes awry is shown in the Wason Selection Task (also known as the Wason 4-Card Task). Before getting too detailed, though, let’s try it. The goal is to solve this puzzle:

If there is an odd number on one side of the card,
then there is a vowel on the opposite side

Which cards do we need to flip over to verify the rule?

Think you have an idea? Upwards of 90% of adults get this puzzle wrong.

The Gambler's Fallacy

At the roulette table, black comes up six times in a row. The gambler thinks, “Since half the time black comes up and half the time red comes up, the wheel is due to land on red,” and bets on red. This is the gambler’s fallacy.

Why is the gambler wrong? Isn’t it true that half the time the wheel will come up black and half the time it will come up red? These are some loaded questions. It is true that given an infinite number of spins of the roulette wheel, red and black will each come up half the time. Even in only a few spins we expect the wheel to come up red and black relatively equally. However, this certainly doesn’t happen every time. At any given point in time, it just as likely to spin a red as it is to spin a black, regardless of how the previous turns came up. That is, the probability function is memoryless.

So why is it that people expect the wheel to ‘even itself out’?

Quantitative Virtual Reality

Theory and practice make for an interesting relationship. The world would be beautiful if all theory came out true in reality. Of course, some things would be difficult to observe in reality. Can you imagine watching in real-time someone shooting a basketball 1,000 times? Or the number of red beetles that drive by? Just simulate it! Seemingly anything of numerical quality can be simulated; this includes basketball shots, options pricing, and the temperatures of the day. This is called a Monte Carlo simulation when a numerical algorithm is run based off repeated sampling of random numbers. 

Base-Rate Neglect

Here’s a probability problem:

At a university, 15% of students are of legal age to drink (21 or older), leaving 85% illegal. There is one liquor store near the campus. Willis, the owner is very good at detecting a fake ID. 90% of the time he correctly stops a student trying to use a fraud license. Unfortunately, Willis isn’t perfect. He also misidentifies proper ID’s 10% of the time. So he has always has an accuracy of 90% when examining students’ identification. Say a student walks in and hands his ID to Willis, and Willis turns it down. What is the likelihood that this student is under age?

The Hot Hand: Truth or Fiction?

Perception is everything. Whether it’s sports, life, or anything else, the way one perceives things changes his or her thought process. This is especially evident in basketball. Imagine that you’re huddled around your players preparing for a buzzer beater. You have a 55% shooter who has had an off-night, only shooting 1 for 10. On the other hand, you have a 35% shooter shooting an extremely well 8 for 11. Now, the last shot of the game: who do you give it to? Before we tackle that, suppose I tell you that I have a fair coin and an unfair coin that is twice as likely to give heads.

Here is a sampling of flips:

H H T T H T H T H H H H H

Would you say it was the fair coin or the unfair coin? If you said the fair coin, then congratulations, you may do well as a clutch-time basketball coach. Because there are twice as many heads up there, many people perceive the unfair coin as being displayed. However, it’s all part of the randomness of a fair coin.
Now, back to the basketball question.

Disease Screening Fallacy

With modern medicine, there remain very few diseases that are a virtual death sentence. However, the fear of terminal illness still lurks in the shadows. How prominent is breast cancer among women ages 18-25? How likelyis it that Huntington’s will be passed from parent to child? Can Diabetes really be prevented? These are all questions that float around the medical world. But do we ever question the diagnostic tests? We certainly ask for a second opinion, but when those test results come in that say “positive”, do we question it? Maybe we should.

Every screening test has it’s own strength ratings (i.e. how good it is at detecting a disease). These are known as sensitivity and specificity. Sensitivity is the probability that a person who has the disease will test positive (or 1- P(false negative)). Specificity measures the rate at which a person without the disease will test negative (or 1-P(false positive)). For many screening tests, these parameters are extremely high. Does that mean we can trust any outcome we get? Not quite. We tend to disregard the original base-rate of these diseases when evaluating the likelihood of testing positive with the screening test. This is better illustrated through an example.

Conjunction Fallacy

When probability roams the free world, it tends to trip people up. One of its prime offenses is the conjunction fallacy. Take the problem:

Gregory is a very accomplished man. He graduated from MIT with a PhD in Computer Science in only 2 years, and started his own software company. He is now a multi-millionaire.

Which of these explanations is more likely:

(A) Gregory is 19 years old

(B) Gregory is 19 years old and a child prodigy

Many people respond that B is a more likely explanation. Let’s try another:

Victoria has spent a lot of time in the hospital. She has undergone 14 surgeries and suffered from 3 heart attacks. She has also been a keynote speaker for the American Heart Association.

Which of these explanations is more likely:

(C) Victoria is an avid baseball fan

(D) Victoria is an avid baseball fan, but suffers from a congenital heart condition

In this problem, many people choose answer D. The correct answers are A and C. Why, do you ask? 

Simpson's Paradox

Ever seen statistics reverse themselves? Try this on for size:

In 2002 Mo Vaughn batted a .259 batting average for the New York Mets, while Raul Gonzalez batted a .260 (for the New York Mets and the Cincinnati Reds). In 2003 Vaughn delivered a .190 batting average, and Gonzalez batted a .230. Clearly in each year, Gonzalez had a higher batting average than Vaughn. However, between the years 2002 and 2003 seasons, Vaughn had a batting average of .258, while Gonzalez only kept up a .240. How is it possible that across two seasons, Mo Vaughn had a higher batting average? Simpson’s paradox.

Tiger Woods: Blessing or Curse?

In 2008 Tiger Woods underwent surgery on his left knee, thereby producing an 8-month layoff from Golf. His departure from Golf caused many golf enthusiasts to stop watching and some analysts even declared Golf too boring without Tiger. His return in 2009 was stated as “one of the most anticipated returns in sports”. During that time, the media machine began picking up around Tiger. First, ESPN ran a commercial with Tiger entering the clubhouse after his hiatus and the tagline, The Party’s Over. A few months later, Esquire Magazine ran a feature article chocked full of awe. I quote:

“Make no mistake: Tiger is different from the rest. I suspect that he sees colors the rest of us don’t know about, that he senses patterns of heat in the grass, electromagnetism in the earth at his feet. The guy bends the world to his will. This makes him feel alien and repellent to the rest of us.”

MCG Signs Multi-Year Deal with ESPN

Providence, Rhode Island based MCG, LLC signed a multi-year deal today with international sports entertainment firm ESPN for the creation of a March Madness Bracket Optimizer for ESPN.com’s popular Insider program. MCG is a mathematical, statistical and strategic consulting firm that specializes in assisting firms with complex quantitative problems. The firm’s unique blend of quantitative expertise and business acumen has allowed them to work with some of the largest firms in the nation.

ESPN, the worldwide leader in sports, is an international sports and entertainment firm owned by The Walt Disney Company.