Why Do I Need to Know Probability and Statistics?

One of the most revealing results from psychology is that we are intuitively horrible statisticians.  In this article, let's explore how poorly we calculate likelihoods of events and why we could derive some benefit from knowing how probability works.

1) You go to see a doctor and she runs a test to see if you have cancer.  If you have cancer, there is a 95% chance that your result will be positive.  If you don't, there is only a 2% chance that your result will be positive.  You are shocked to learn that the result was positive.  How worried should you be?

If you're like most people, you would be terrified because you'd think that the odds of you having cancer is about 95%.  This is a classic example of ignoring the base rate which in this case is the percentage of people in the U.S. who get cancer each year.  It turns out that percentage is roughly 0.5%.  Given this information, you can calculate how likely it is for you to get a positive test result whether you do or don't have cancer.  Despite getting the positive result, you have a probability less than 19.3% of actually having cancer (after applying Bayes' Theorem from elementary probability theory).

2) Here's a simpler one that is deceptively tricky.  There's a couple who has two children and you are told that one of them is a girl.  What are the odds that the couple has two daughters?  Again, most people think 50% since the other child has a 50/50 chance of being male or female.

Let's go through this problem step by step.  Note that there are four possibilities:

OLDER CHILD   YOUNGER CHILD

Girl                        Girl

Girl                        Boy

Boy                        Girl

Boy                        Boy

You know that they have at least one girl so you must get rid of the last possibility; all the others are fair game and they all have equal likelihoods.  Thus, the probability of having two daughters is actually only one-third, not one-half.

3) Ashley majored in English literature at a small liberal arts school where she was active in theater, music, and political activism concerning women's and minorities' rights in America.  It has been over a year since she graduated from college.  Given five possible post-graduation outcomes, place a probability next to each possibility:

A) She is pursuing her dream to become an actress.

B) She is working as a banker.

C) She is working as a banker and auditioning for acting roles at night.

D) She is attending law school.

E) She is unemployed.

Given what you know about Ashley, take the time to actually place a probability for each outcome.  

Okay, done?  Great!  First, make sure that your numbers don't add up to over 100% since that wouldn't make any sense.  Second, what probability did you assign to B?  How does that compare to the probability you assigned to C?  Many people would assign C with a higher probability than to B since Ashley seems like she's very artistically inclined.  However, this is actually impossible.  The probability of her working as a banker AND auditioning for acting roles must be lower than the probability of her working as a banker.  One way to think about it is that the probability of her working as a banker is equal to the probability of her working as a banker and auditioning for roles PLUS the probability of her working as a banker and NOT auditioning for roles.  Thus, the probability of C cannot be greater than the probability of B.

These types of puzzles unveil some of the imperfections of our "rational" minds.  If you are interested in exploring the faulty statistics that go on in our heads, I highly recommend Daniel Kahneman's Thinking: Fast and Slow and Leonard Mladinow's The Drunkard's Walk: How Randomness Rules Our Lives.  I may also post more about these issues in the future.

10 Tips for a Healthier and Fitter Body

After a year and a half of getting out of shape since college graduation, I’ve decided to try to improve my physique and overall health.  During my fitness research, I have encountered several tips that immediately gave me some good results.  I thought that I would share them with you in case you are interested in improving your diet and lifestyle.

1. Avoid fruit juices.  They have a high sugar to fiber ratio that’ll stress out your pancreas from the spike in blood sugar levels.  Also, this will lead to unnecessary levels of fat storage.  Eat fruit instead!

2. Drink less alcohol.  You are probably already aware of this fact but it is true.  Alcoholic beverages are not only calorically dense (hence the beer belly) but are completely void of nutritional value.  If you must, go with light beer or red wine.

3. Consume complex carbohydrates and avoid simple ones.  White bread and white rice are broken down in your body very quickly and lack all the nutrients whole wheat bread and brown rice have.

4. Eat breakfast.  In fact, get into the habit of eating something within the first hour of getting out of bed.  It will kick-start your day’s metabolism and provide you with the energy to engage in your daily activities.  According to Men’s Health magazine, most overweight people skip breakfast.

5. Drink lots of water.   Two-thirds of your body is made up of water so you need a good amount to make sure your body is functioning properly.  A good benefit of being well hydrated is that your appetite will be stable, preventing you from eating unnecessarily.

6. Eat more meals throughout the day.  This one may be counterintuitive so let me explain.  If you only eat one or two meals a day, your metabolism will slow down because you’re body will want to store energy (thus fat) to last you through those long intervals without any food.  If you eat five or six meals a day (eating every 3 hours or so), then your metabolism will keep going throughout the day, burning more calories.  This is a good way to increase your protein intake if you want to add more muscle.

7. Make sure to eat healthy fats.  Your body needs fat to function properly but it is important to keep track of the fat sources.  A general rule of thumb: get your fats from plants (e.g. canola, olives, beans) and fish.  Avoid the fats from dairy and red meat.

8. Stay away from soda.  The diet alternatives are still bad because of the high sugar content.  Sodas have high levels of phosphorous which results in calcium loss, increasing the risk of osteoporosis (i.e. bone density loss).

9. Stop drinking coffee.  Your body should not need caffeine to get you energized in the morning or at any point during the day.  If you feel tired, then that’s your body’s way of telling you to rest.  Dumping coffee into your system is not a healthy way to solve the problem.  If you are addicted, you may experience withdrawal; try to slowly wean yourself off of it.

10. Eat plenty of plants (fruit and vegetables).  If you look at our dental structure, it is more herbivorous than carnivorous; most of our teeth are for cutting and grinding plants.  Evolutionarily speaking, our bodies are adapted to sub-Saharan African jungles and the available food sources there.  A major problem with our diet these days is that we now eat foods that did not exist or were rare in our prehistoric environment.  This is why obesity and diabetes have become so rampant.  Sugar was scarce but provided a lot of energy; thus we developed sweetness to detect sugar, which then inclined us to consume more of it.  Now, sugar is ubiquitous and many devour it carelessly to their demise.

So there you have it.  I have tried to avoid mentioning the obviously unhealthy habits (e.g. smoking tobacco, snorting cocaine, eating fried food, shooting up heroin); if you do any of this already on a regular basis, I highly encourage you to stop.