Simulated experience #1

In a random group of 25 people, what would be the probability that there are 2 people with the same birthday?

This is the famous birthday problem.

We have difficulty in understanding the probabilistic structure of this problem. When I asked this question to 100 university students, the average response I got was “less than 1%.”

The correct answer is actually around 54%!

The analytic solution of the problem is not straightforward. One has to think in terms of combinations. One has to make several complex calculations.

And these don’t come naturally to us.

So what to do?

If you think about it, experiencing frequencies of outcomes would be actually much easier. This would mean going out there and meeting many many groups of 25 people and observing if there are matching birthdays in each group.

Sounds like a hard task to accomplish though.

But wait; we could use simulations.

Check THIS site for instance.

What you do here is to generate 25 birth dates and see if there are two that are the same.

Then reset and meet a new group of 25. Then again. Then again. Then again…

Once you meet a dozen groups like this, you’ll notice that in around half of them, in fact, there is a match.

Consider now the power of simulations and the possibilities of experiencing the outcomes of complex and relevant probabilistic scenarios, such as investment decisions, pension schemes, insurance regimes, with the help of simulations. You would gather much more insight on the problem and possibly make better decisions.

Well… Try with a number larger than 25 this time.

You’ll be surprised.

Transparency

Transparency of a description denotes how correctly it is perceived and accurately understood by people. If I tell you that, as a side effect, the use of a certain drug increases the probability of say, becoming completely bold by 100%, you would have second thoughts about even touching the pill. But you are missing a crucial piece of information there: the base on which this statistic was calculated. For instance, an increase from one in a million to two in a million would also constitute a 100% increase. The drug seems less scary now, doesn’t it? Hence, the description that includes the base rates is more transparent to the human mind in this case.

Gerd Gigerenzer, Wolfgang Gaissmaier, Elke Kurz-Milcke, Lisa M. Schwartz and Steven Woloshin, in their 2007 report published in Psychological Science for Public Interest (downloadable here), dug deeper into the issue. At one point, they talk about why abortions in England and in Wales increased dramatically around 1995. The reason was that the birth control pills were rumored to have an undesirable side effect, expressed in a non-transparent way, which created an overreaction against their use. Ironically, it was found that the abortion procedure increases even further the probability of facing the same side effect, hence the importance of transparency of a description, especially in medicine.

The most curious cases are medical tests. They can make two types of mistakes: finding a sick guy healthy or a healthy guy sick. They are commonly designed to avoid more the former one (finding a sick guy healthy). On the other hand, one can observe more cases where they find a healthy person sick, especially if there are much more healthy people than sick ones. Consider a test that always correctly identifies a sick person and fails to find a disease (correctly) in a healthy person 90% of the time, i.e. 10% of the time it erroneously claims that a healthy guy is sick. Say that the probability that a random person in the population has the disease is 1%. We grab a random guy from the population and test him. The test says he has the disease. What is the probability that he is really sick then?

Answer: Take the test again.