Wednesday, May 15, 2013

What we should know about probability


The concept of probability is often used but rarely properly understood. The first thing we need to understand about probability is that it is a human construct. This means probability is something we made up. We made it up to describe our ignorance and our beliefs about the world. Since our beliefs are, by definition, always subjective, the concept of probability is subjective as well. Another important thing we need to understand is that probabilities are useful in repetitive events, but they tell us nothing about single, unique events.

Just recently, one famous Hollywood actress decided to undergo a surgery to remove both of her breasts because the doctors found that she "had an 80% probability" of developing breast cancer due to the presence of a certain gene. Without the intention of judging her choice, let us examine what this 80% probability means. It means that 80% of the people examined in the past that had this particular gene and that followed certain lifestyle had developed breast cancer. For those remaining 20%, researchers could not find a distinguishing feature, compared to the other 80%. In other words, if there are 100 individuals with this particular gene, and these individuals have similar lifestyles (including all the environmental conditions surrounding them), 80 of them will develop breast cancer and 20 won't.

We are tempted to say that an individual from this group has an 80% probability of developing breast cancer, but this is incorrect. If you look at an individual that had developed breast cancer and ask: can I use the 80-20 ratio to explain why this particular individual had developed breast cancer, the answer is--no. You can ask the same question about an individual that did not develop breast cancer and the ratio would be equally uninformative. It wouldn't tell you why individual X or Y ended up in the 80% group and individuals Q and Z ended up in the 20% group.

Let's say this particular actress is actually in the 20% group, but, of course, no one can tell her this because no one knows this underlying reality. This means that she was in this group before she learned about the 80-20 ratio, and she is in the same group now, after she learned about the ratio. She still doesn't know in which group she is. So, learning about the 80-20 ratio describing the health of some other people did not add any new information about her health.

What do probabilities really mean then? The real meaning of probabilities is that they are our own subjective expression of our beliefs about the unknown. We base these beliefs on our evaluation of past experiences. Our famous actress has observed the past results of medical research and she formed certain beliefs about her health, which is still unknown to her. We don't know her actual thought process, but her actions demonstrate that she would rather remove her breasts and reduce her subjective belief that she is in the 80% group than wait and see in which group she actually is. 

But, suppose that, instead of the 80-20 ratio, this ratio was 99-1. This means that of every 100 individuals with the particular gene and similar living conditions, 99 get breast cancer, and 1 doesn’t. Suppose you were that one person that will not develop cancer, but, of course, you don't know this. The doctor comes to you and says the chances of not getting cancer for you are 1%. What does that number tell you about your health? Nothing.

This is because we are dealing with single, unique events. Only if you had the opportunity to live your life 100 times, could we say that you will not develop cancer only in one of those 100 lives. But, you are living only one life, and that’s why applying probabilities to a life of a particular person is inappropriate.

Similarly, as far as we know, our actress is not going to relive her life 100 times. This is why using the terms like “she had an 80% chance of developing cancer” are inappropriate. The 80-20 ratio is no more informative about this actress’s underlying objective reality than any other arbitrarily chosen ratio. The number is, however, quite informative about her subjective reality--her beliefs.

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