We received great private feedback
on our last web log article, Aristocrats in flyovers, including
from some extraordinary business and public sector colleagues and readers. It is worth taking a second look at the
probability math underlying that article, so that way it's easier for you to make sense of any related paradigms that you heuristically experience. In this follow-up note, we explore
a simple probability exercise and reflect on the impact of luck versus skill, combined with
sample size, in explaining the extreme concept of uber-wealth.
Keep the feedback coming, and using the comment feature
below is the easiest way to do that.
Our exercise here looks at 2 teams of monkeys: Team NY with 16 monkeys, and Team NE with 4 monkeys. Don’t worry about the team names for the time being. The teams are identical in every way, but for the number of monkeys in them. All 20 monkeys started with $0 wealth, and they are each given 1 coin and toss the coin 4 times. For each coin toss landing on heads, the monkey receives $1.
There is obviously no skill involved with monkeys successfully tossing a coin. So at the end of the exercise shouldn't the expected wealth of Team NY's richest monkey, and the expected wealth of Team NE's richest monkey, be the same? The point of this article is to reinforce for you that concept that we must expect different outcomes, in a pre-defined way, from chance alone!
Here is the wealth probability distribution that each monkey has through the 4 coin tosses:
Our exercise here looks at 2 teams of monkeys: Team NY with 16 monkeys, and Team NE with 4 monkeys. Don’t worry about the team names for the time being. The teams are identical in every way, but for the number of monkeys in them. All 20 monkeys started with $0 wealth, and they are each given 1 coin and toss the coin 4 times. For each coin toss landing on heads, the monkey receives $1.
There is obviously no skill involved with monkeys successfully tossing a coin. So at the end of the exercise shouldn't the expected wealth of Team NY's richest monkey, and the expected wealth of Team NE's richest monkey, be the same? The point of this article is to reinforce for you that concept that we must expect different outcomes, in a pre-defined way, from chance alone!
Here is the wealth probability distribution that each monkey has through the 4 coin tosses:
$0 1/16 *
$1 4/16 ****
$2 6/16 ******
$3 4/16 ****
$4 1/16 *
An average wealth of $2, with the potential for $4. Now let’s explore
the expected wealth of the richest monkey in the two teams, to see if how they compare.
Team NY (sample size 16)
Team NY (sample size 16)
The probability
of a monkey amassing $4 through sheer luck is 2^-4, or 1/16. So given Team NY has 16 monkeys, can we expect Team NY's richest monkey to have an expected wealth of the highest possible value, or $4? No, this would be wrong (though not by much).
Instead the expected wealth of Team NY's richest monkey is a little less than $4 (e.g., about 15/16 of $4), and this is just the beginning of what's important to learn here. Certainly the 1/16 probability of each monkey amassing $4 is just the typical path. But the actual random distribution of the number of Team NY monkeys amassing $4 will sometimes vary from 1 monkey. It could in some cases be more than 1 monkey (in which case the richest monkey would just still tie at $4 in wealth). But in other cases it would likely resolve with 0 monkeys amassing $4 in wealth, and in most of those cases the richest monkey would instead have $3 of wealth (see the probability illustration above to imagine this better).
It turns out from the probability mathematics model that the expected wealth of Team NY's richest monkey is not $4, but instead a little lower at $3.7.
Team NE (sample size 4)
Instead the expected wealth of Team NY's richest monkey is a little less than $4 (e.g., about 15/16 of $4), and this is just the beginning of what's important to learn here. Certainly the 1/16 probability of each monkey amassing $4 is just the typical path. But the actual random distribution of the number of Team NY monkeys amassing $4 will sometimes vary from 1 monkey. It could in some cases be more than 1 monkey (in which case the richest monkey would just still tie at $4 in wealth). But in other cases it would likely resolve with 0 monkeys amassing $4 in wealth, and in most of those cases the richest monkey would instead have $3 of wealth (see the probability illustration above to imagine this better).
It turns out from the probability mathematics model that the expected wealth of Team NY's richest monkey is not $4, but instead a little lower at $3.7.
Team NE (sample size 4)
Now clearly with only
4 monkeys and a slim 1/16 chance of any one monkey amassing the highest $4 wealth potential,
the richest monkey here is more likely to have a wealth tied closer to $3 instead. The $3 level accommodates the highest
4 of the 16 (or highest 1/4 of the) combinations shown in the probability illustration above.
Here too we might fall into the quick trap of miscalculating the expected wealth of Team NE's richest monkey to be:
Here too we might fall into the quick trap of miscalculating the expected wealth of Team NE's richest monkey to be:
$4*1/16 + $3*3/16 = $3.25
We would be directionally correct
in thinking here that Team NE's richest monkey has a lower wealth than that of Team NY's richest monkey (3.25<3.7<4). And that's an interesting observation given that all of these 20 monkeys are identical, and all without skills. The only reason we see the pre-defined wealth difference is from the difference in sample size (there is no other difference at all).
But the $3.25 guess for Team NE's richest monkey is incorrect and it is subject to the same downward modification from convolution (use search field above for many more web log articles on this probability concept) as we saw for Team NY. As (less) luck would have it, the richest monkey is expected to have wealth a little less than $3.25 (e.g., about 3/4 of $4). And again the original $3.25 guess was merely the typical probability path.
At this point if the mathematics is difficult, please skip to the final section highlighted "Final remarks". It doesn't make sense to miss out on the main story, simply because of complexities within probability theory here. If not, then please continue reading everything else below.
To hone in only on the weakest part of that probability guess though, we can state that the probability that Team NE does not have any monkeys amassing $3 -specifically- is about:
But the $3.25 guess for Team NE's richest monkey is incorrect and it is subject to the same downward modification from convolution (use search field above for many more web log articles on this probability concept) as we saw for Team NY. As (less) luck would have it, the richest monkey is expected to have wealth a little less than $3.25 (e.g., about 3/4 of $4). And again the original $3.25 guess was merely the typical probability path.
At this point if the mathematics is difficult, please skip to the final section highlighted "Final remarks". It doesn't make sense to miss out on the main story, simply because of complexities within probability theory here. If not, then please continue reading everything else below.
To hone in only on the weakest part of that probability guess though, we can state that the probability that Team NE does not have any monkeys amassing $3 -specifically- is about:
[p(1 monkey not having $3)]^4
= [12/16]^4
~ 5/16
So in our
convolution model, for the 3/4 or so times that Team NE's richest monkey does not
actually amass $4 in wealth, a good portion of those times all monkeys (including Team NE's richest) will not have amassed $3 either. One may think 15/16 and 3/4 in this article seems like some variation of the (n-1)/n correction we could make on variance, but this is simply an incorrect oversimplification as we are conflating a variance (deviations^2) correction as if it were a linear relation.
Now, sure, some monkeys may tie at $3. But the main point here is that the expected wealth of the richest monkey otherwise falls instead to the $2 unconditional average.
We see from the probability mathematics here that expected wealth of Team NE's richest monkey is not $3.25, but instead a little lower at $3 (a full dollar less than $4).
Final remarks
Now, sure, some monkeys may tie at $3. But the main point here is that the expected wealth of the richest monkey otherwise falls instead to the $2 unconditional average.
We see from the probability mathematics here that expected wealth of Team NE's richest monkey is not $3.25, but instead a little lower at $3 (a full dollar less than $4).
Final remarks
In an experiment
where monkeys could average a net worth of $2, having a net worth of $3 or greater
is of course a minority outcome and one that can many times happen through chance alone (by definition you can't call all monkeys tossing coins "skilled".) This example was meant as a loose disguise to the modeling
paradigm we employed in our previous web log article. Team NY could
represent the populous of New York; Team NE could represent the more modest population of Nebraska.
And as a result we can now expect in a pre-defined way the richest person in New York to be wealthier than the richest person in the flyover of Nebraska, simply from sample size alone and nothing else related to underlying economics! Being the richest person in a team or state is something we also know is easier when competing in a smaller population. In the case of equal monkeys, one has a 1/16 probability of be being the richest in Team NY but has a 4/16 (or 1/4) chance of being the richest if in Team NE. And meanwhile the relative worth of those richest is fairly nice (e.g., $3, when the average was $2).
And as a result we can now expect in a pre-defined way the richest person in New York to be wealthier than the richest person in the flyover of Nebraska, simply from sample size alone and nothing else related to underlying economics! Being the richest person in a team or state is something we also know is easier when competing in a smaller population. In the case of equal monkeys, one has a 1/16 probability of be being the richest in Team NY but has a 4/16 (or 1/4) chance of being the richest if in Team NE. And meanwhile the relative worth of those richest is fairly nice (e.g., $3, when the average was $2).
Let's now revisit this chart on the left (click on illustration to zoom in). We see we are
at the cusp here with the richest individuals among us, explaining their wealth as
a matter of chance. Or for some, as a matter of relative mediocrity being propelled by forces beyond those individual's control. The gold data bars shows
the equivalent of what wealth we might expect of the "richest monkey" from that state team. And the mostly red (there are two blue) data bars show instead what wealth has actually been amassed.
What's interesting in this geographic richest list is to see someone such as Warren Buffett (aged 84), from the less populated area of Nebraska, propel his wealth to consistently be well ahead of -for example- the richest person in New York. Michael Bloomberg (aged 72), whose office I had presented research to a few years ago, currently tops the rich list for the large state of New York, but isn't as consistent a long term performer in these ranks as Warren Buffett is. This type of outperformance by Warren Buffett indicates a much greater tilt of explanation towards wealth-generating skill versus luck, when contrasted with most other people anywhere on the list.
On a final, unrelated note, happy to share that outside the U.S., the Statistics Topics book has been within top 2 in probability and statistics (at times beating out Levitt's Freakonomics and Silver's Signal & the Noise). Not only has the book been top ranked also in mathematics research for each month so far of Q3 and Q4, but half of this year's book income is being donated to charity.
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