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.”