”Rub of the Green“
Tel Aviv
05/01/2012 00:00:00
I confess, I detest brainteasers. Even if I may have indulged in their solution in my youth – at times with great success – I lost all interest in doing so past the age of fifty. Since I also purged from my life other forms of unproductive entertainment such as crossword puzzles and computer games, I generally assume that anyone with fewer years ahead of him than behind him, becomes more judicious with how he spends his time. At any rate, the mere possibility of my eight-year-old grandchild solving a puzzle that I can’t is enough to dissuade anyone from attempting amusement by puzzle-solving.
It’s a pity, I thought recently, when I came upon one exceptionally perplexing brainteaser. I then realized that when we give up on a challenge, like this one for example, we deprive ourselves of important insights about the dynamics of our decision-making process.
Think for a moment about the following puzzle. There are five envelopes before you. In each envelope one colored card is concealed, drawn at random from a pile of one hundred cards that consists of 36 green cards, 25 blue cards, 22 yellow cards and 17 red cards. You are asked to guess the color of the card in each of the five envelopes.
Now, most people would base their answer on some form of mathematical distribution. They would say green in nearly one third of the times, and assign the colors of the other cards inside the envelopes in accordance with their relative frequency in the pile. With slight modifications, this would be a relatively common approach to solving the puzzle. And yet the true solution is different. For if we were to examine each envelope individually, we would see that we must wager that each one contains a green card, as that color appears most frequently.
This solution is aggravating, among other reasons, because it requires that we forfeit, in advance, the chance to correctly guess the colors of all the cards in the envelopes, and accept up front that we will probably miss on two-thirds of our guesses.
The behavioral bias that guides roughly ninety percent of those attempting to solve this puzzle is called “Probability Matching”, and has become a familiar subject in the study of the decision-making process under conditions of uncertainty.
If we were asked about the color distribution of the cards of all envelopes, as a group, the answer would of course correspond to the relative frequency of the color in the pile of cards. But when each envelope has to be evaluated independently of the others, the right guess would be that the card inside it would be of the most frequently-appearing color, green in our case. The reason that this approach is optimal stems from the fact that statistically, it offers the smallest margin of error. Similarly, if we had to guess the weight of a random pupil in a classroom, the best answer would statistically be the average weight of all students, because again, it offers the smallest margin of error.
Among cognitive psychologists, one widespread theory that seeks to explain the popular bias, stipulates that there are two parallel systems operating in our minds, alternating according to need. One system is intuitive, quickly processing large quantities of information in an associative manner, and is responsible for what we generally call “gut feeling”. This system is our brain’s default choice, and is responsible, among other things, to cope with existential threats. It has many advantages, but one of its noteworthy disadvantages is its inability to process complex probability algorithms. The system in charge of this function is our rational system: the controlled, the analytic, the calculative. It operates with much effort and according to predetermined laws. Now, the correct solution to the above puzzle could be reached using our rational system. However, our intuitive system is twice as fast, and is quick at offering its own solution, in this case the wrong one, of matching up card colors by general probability.
Lab researchers, in an attempt to trigger the rational system to operate in test subjects, use the following method. They tell them that this is a statistical test (as opposed to random lot), and that there is no time constraint. These are some of the conditions that trigger the operation of the rational system. And indeed, a significant proportion (43%) of the subjects that were instructed in this way was able to suppress the quick intuitive response, and solved the puzzle correctly.
Yet in my opinion, the important lesson of this puzzle is the one about the winning statistical approach of minimum mistakes. It is true that a wild guess can yield a better result, but whether you need to guess colors in the envelopes or make life choices, making fewer mistakes is often the winning strategy.