Numinations ó July, 1999


© 1999, by Gary D. Campbell

Have you ever received a chain letter? Suppose you did. Consider a typical chain letter. It could be started by a person sending out two letters with his name at the bottom of a 20 person list. You would be asked to send a dollar to the person at the top of the list, remove that person from the list, add your name to the bottom, and send two copies of the revised letter to two other persons. Each time the letter was received and the instructions followed, there would be twice as many recipients. There would be approximately a thousand letters after ten generations, a million after twenty, and a billion after thirty (the number goes up by a factor of 1024 for each ten generations). The person initiating the chain could expect the 19 friends he listed above himself to share over a million dollars (two dollars for the first person, four dollars for the next, and so on). He would expect to receive a cool million for himself.

Chain letters are illegal. They are like pyramid schemes, which are also illegal. These scams involve large gains for the first few people to become involved, and large losses for the many people who follow them. Our government has decided that itís bad for us to play such games, although they have devised the Social Security system, which works (or fails to work) in much the same way. Letís Numinate about the very first chain letter that must have been sent many years ago. Assume that all recipients participated. Imagine that you were one of them. The question is, how much could you have expected to receive if you followed the instructions?

Letís assume this was back when the worldís population was one billion people. You first note that twenty names are on the list. You figure how it grows and conclude that the game will be over after at most 30 generations. You can calculate that the most the person originating the chain, or anyone else, could receive would be about $1 million. It should be obvious to you that the later people into the game might receive nothing. Their names canít be promoted to the top, because there are no new people to send letters to. The final few generations could consist of letters to people who had already played the game, and who knows how many times they would play it over. But, letís go with the simpler assumption that everyone plays exactly once. This means a billion dollars are paid into the game ($1 for each person on earth).

How is the money paid out? The persons the originator listed above himself will share about a million dollars among them. The originator and the people in the nine generations that follow him will share the remainder of the money (for a total of a billion dollars). There are 1023 such individuals. So, it appears that a few more than a thousand people will share about a million dollars each. A thousand winners out of a total population of a billion means that any person chosen at random would have about a one in a million chance of being one of the winners.

Are the true odds of being a winner one in a million? This is a sequential phenomenon. It would seem that some additional information might pertain. For example, it might make sense to assume that about half the population received their letters before you did, and half will receive their letters later. Depending on the vagaries of mail delivery, that means you are in the 29th or 30th generation. In any case, assuming you are anywhere close to the middle person receiving a letter, you must be among one of the last two generations of the chain. If so, you have no chance at all of being a winner (of course, zero is not much less than one in a million).

You could make a more optimistic assumption: That your name was added to the list about half way through the game, namely that you are in the 15th generation. But, again you are at least five generations too late; only the originator and the nine generations that follow him will receive a million dollars. It takes the first 20 generations to pay the 20 people on the original list. The originator, himself, is paid by the 20th generation. The 21st through 29th generations are sending their dollars to the lucky people in the first nine generations following the originator. The game begins dying off starting with the 30th generation, if it hasnít already done so before that time, because the 30th generation would require the entire population of the world to send two billion more letters. Now, with each generation, every person would begin receiving two, then four, then eight letters apiece. You would think they would catch on at this point and drop out of the game. No one after the 10th generation can get promoted all the way to the top of the list. They simply have no chance at all.

If you understand this analysis, there is a natural extension of its logic. It has been called the Carter/Leslie doomsday argument. It runs something like this. Here you are, a living human being. There will ultimately have been some number of human beings before the mold is broken and our species goes extinct. You may be among the first, the last, or somewhere in the middle of the line. All things being equal, it is reasonable to assume that you were born somewhere in the middle. However, like the chain letter, there are more humans with each generation. About ten percent of all humans ever born are alive today. If humans were to continue reproducing at their current rate for another hundred years, about half would have preceded those alive today, and about half would follow us. If the rate could continue for another millenium, you and I would be among the first 0.001 percent of all humans to have lived. What are the odds? The Carter/Leslie argument is that the odds are that you are more likely near the middle. This can only be the case if the current rate of human reproduction stops (or almost stops) in a hundred years or so. The conclusion predicted is what they call ďDoomsday Soon.Ē No reason for Doomsday needs to be given. The case is purely mathematical.

Although this particular application is fairly new and not very well-known, arguments like this are common in science. There are basically three kinds of these arguments. The first, and most well known, is Occamís razoróthe proposition that the simpler of two competing explanations is the preferable (or more likely). The second is that given a single data point (such as when you were born), assume that it lies near any relevant mean (such as halfway in time from First Man to Last Man). However, when faced with extremely unlikely circumstances, the third option may be chosen. This was titled the Anthropic Principle (by Carter of the Carter/Leslie, above). It basically explains why we might find ourselves on a planet that might be one in a thousand, orbiting a star that might again be one in a thousand, perhaps in a galaxy thatís also one in a thousand, and in a universe with laws of physics that are unlikely beyond estimation. The reason is simple: Unlikely as they are, if all these things hadnít come to pass, we wouldnít be here to observe them. Our being here changes the odds according to the rules of conditional probability.

The doomsday argument is based on Bayes law. Suppose you were shown two urns, and told that one urn contains 1000 balls and the other contains only three balls. Furthermore, each urn contains exactly one black ball. Now, someone reaches into one of the urns and draws out a black ball. Your problem is to pick the urn that contains the fewer balls. There are two urns, so itís a 50/50 choice, isnít it? Or, is it more likely that the black ball was drawn from the urn that contains only three balls? How much more likely? Given the truth of what you know and have seen, Bayes law tells us thereís a 99.7% chance that the urn from which the black ball was drawn has only two balls remaining in it.

The doomsday argument asks you to pick the more likely scenario: You were born among the first 0.001 per cent of human beings, or you were born at about the 50% point. If you think the odds favor the second hypothesis over the first, then you have to think that the odds also favor Doomsday Soon, especially given the growth curve of world population. In any case, itís worth Numinating about, donít you think?

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