Recently I posted articles about gambling on a three-dice throw and whether children are boys or girls. The point was to emphasize that making good decisions requires knowing which events are correlated. This is most obvious in the fights that still occasionally erupt over the Monty Hall paradox. [In brief: You have three doors. One hides a treasure and two have nothing of value. You choose a door. Monty opens one of the other doors and shows nothing is there. He then asks if you want to keep your choice or change. Many people find it hard to believe you double your chances of winning if you switch.]
These puzzles are important in real life. Failure to analyze correctly the correlations between parameters in the housing market resulted in the collapse of our economic system (greed and other factors such herd mentality allowed this error to happen). For a very readable explanation, see the article called “Recipe for Disaster” by Felix Salmon.
Without equations or graphs, the article builds the case that misuse of a brilliant piece of mathematical work cost trillions of dollars. One example I like is his consideration of kids in grade school. Assume the probability the parents of a child get divorced is 5%; the probability of seeing the teacher slip and fall is 5%; the probability of having head lice is 5%; and the probability of winning a spelling bee is 5%. Two children sit next to each other. If one’s parents are divorced, the probability the other’s parents are divorced is still 5% (to our accuracy assuming the families are uncorrelated), but since they sit next to each other, if one sees the teacher fall, the other child will likely see it also (weak correlation). If one child has lice, they will likely jump and infect the other (a stronger correlation). If one child wins the spelling bee, the other is certain to lose (a strong negative correlation).
Naively a person could say the probability of two children both having divorced parents is 0.05*0.05 = 0.25% and that is a reasonable estimate. However, what is the probability that both children see the teacher fall? Given that one might be looking the other way and miss the event, what can you say? The last example is the easiest. If we ask what is the probability that both won the spelling bee, the answer must be zero unless the rules are changed to allow ties.
These examples sound trivial, but what is the probability that a house will go into foreclosure if another in the neighborhood forecloses? Suppose several houses in a neighborhood go into foreclosure, what does that mean for the remaining ones? These are not isolated events, they are more like the head lice example — if one has it, the probability the other has it jumps a lot.
The actual situation as it unfolded was even worse. New investment instruments were invented with valuation based one the market valuation of other similar things, not on the underlying assets. Combine that level on misunderstanding with greed (private and corporate), and we have a recipe for disaster. Risk is always present in an investment and that is not necessarily a bad thing. Not knowing the actual risk is extremely dangerous.
Do not bet on the three-dice game unless you are willing to take about an 8% loss.