The key to my insight was my friend's reference to a scam. Once personal feelings about scams are set aside, I think studying how scams work can be pretty interesting (and in this case, instructive). One scam I’ve always been fascinated by is the ‘hot stock tip’ scam. In this scam, a scammer collects a huge list of recipients and sends them information about a hot stock tip. The goal is to convince a recipient that the scammer actually has inside information about a stock. Once convinced, the scammer will then ask the victim for money to invest on their behalf. The scam is that instead of investing it, the scammer just keeps the money (and presumably, runs).
Now, how could this ever work if the scammer has no inside information? The catch is the way the list is split up – half the list gets a tip that the stock is going up while the other half gets the opposite information. After this information gets sent out, the scammer waits for the market to move. Once the stock has clearly moved up or down, the scammer sends another tip about a different stock. But the new tip only goes out to those who originally got the tip that correctly predicted what happened in the market. The people who got the incorrect tip are never contacted again.
Do you see what is happening, reader? Over time, the scammer is sending out less and less mail. However, the mail is increasingly convincing because the recipients have been receiving correct predictions about the market for a long time. The likelihood the scam succeeds is based almost entirely on the size of the initial pool – the larger the pool, the more mail that can be sent and the more convincing each subsequent stock tip becomes.
The reason why I thought this scam applied to my friend’s rant is because it uses math to explain how a large enough sample size is a sufficient explanation for how objectively stupid people can amass incredible fortunes. If we have enough stupid (or willing) people, all we need to allow one lucky person a massive streak of good fortune are the laws of basic probability. Let’s use a carefully constructed hypothetical example to illustrate the point.
Suppose you have a sample size of 'n' people at the start of a year. This group has no business acumen and no particular edge in intelligence on others in their society. They live, in other words, almost completely normal lives. However, this group does have a high appetite for risk. Once a year, this entire group goes to a casino and bets their entire savings from the year – let’s call it $100 – on the roulette wheel. Half of these fools bet ‘black’ and the other bet ‘red’ (editor's note for degenerate gamblers: in this hypothetical example, we’ll have to ignore the ‘green’ spaces).
Next year, the same thing happens, but only those who won the prior year come back (the other half swear off gambling for good after losing it all and become highly respected pillars of their respective communities, or bloggers, or whatever). They again bet in a perfect split – half red, half black. The catch is that this time they bet all the money they won last year - $200. As it was the year before, half win and half lose. This cycle repeats itself next year, and the year after, and the year after that…
Do you see what is happening, reader? What this hypothetical demonstrates is that for a given combination of a sample size ‘n’ and the number of years ‘x’, there will always be at least one person who has won every single annual bet. I tried to work out the math and came up with the following formula:
Prize money for the ‘big winner’ = 100 * 2^xThis formula is bound by a restriction because the math only works if n is large enough to guarantee at least one bettor on each color in every year of the scenario:
Prize money for the ‘big winner’ = 100 * 2^x ( while n >= 2^x )
(Please note that ‘>=’ means ‘greater than or equal to’.)(For those still unsure of the math, please see the endnote for a breakdown of how n and x interact in the above scenario.)
If we continue on with this example, we can see how certain combinations of n and x can lead to immense quantities of wealth for the big winner:
n = 100 and x = 6 : $6,400One way to read the last line of the above is to say:
n = 1,000 and x = 9: $51,200
n = 10,000 and x =13: $819,200
‘For a given initial bet of $100, at least one person out of a group of 10,000 will amass almost a million dollars so long as the group’s distribution of all or nothing wagers over a thirteen year period is evenly distributed among two distinct and independent outcomes where each outcome has a 50% chance of occurring.’
This outcome is wordy but I encourage you to try and fully understand it, reader, for its consequences are significant in all manners of applications. First and foremost, it challenges a commonly accepted perception about wealth. Although it is certainly true that an exceptionally intelligent or hard-working individual can amass a great deal of wealth over a lifetime, it also seems to be the case that if the sample size of crazy risk takers is large enough then at least one out of the group will amass his or her own fortune over a prolonged period of risk taking.
Footnotes / endnote
0. OK, so I get all of it… except n and x… so, uh, I guess I DON’T get it…
Let’s build the intuition here by using a simplified example. Suppose you and seven of your college friends decided to do some version of this exercise (n = 8). In year one (x = 1) you all bring your $100 to the casino and bet as stipulated above – half win, half lose.
The next year (x = 2) the four winners return with $200 to try again. Same result…
In year three (x = 3), two winners return with $400 and try again. This time, only one winner will remain. After this third year, the scenario is officially over because there are not enough people remaining to cover both red and black. Thus, it is 50/50 whether any person in the original sample ‘n’ will win big or lose and return to $0.
The formula above for the example of n = 8:
Prize money for the ‘big winner’ = 100 * 2^x ( while n >= 2^x )By year:
x = 1: 100 * 2^1 = 100 * 2 = 200 (while 8 >= 2)As seen above, this example breaks down in year four:
x = 2: 100 * 2^2 = 100 * 4 = 400 (while 8 >= 4)
x = 3: 100 * 2^3 = 100 * 8 = 800 (while 8 >= 8)
x = 4: 100 * 2^4 = 100 * 16 = 1600 (but the condition 8 >= 16 fails)Granted, the condition n >= 2^x does not need to hold for someone to win big. However, since it is impossible to have at least one person from the group wagering on each color, there is always a 50/50 chance of having no winners. I don’t dig too far into this aspect of the scenario because the point of this exercise is to demonstrate how a large enough sample size of risk takers betting arbitrarily guarantees at least one risk taker will emerge with a level of wealth unimaginably greater than the starting amount.
