"The Urn of Mystery"

Here is one more example from Douglas Hubbard's book, "How to Measure Anything: Finding the Value of "Intangibles" in Business" - this one is about something that Hubbard calls the "Urn of Mystery."   As a reminder, Hubbard claims that small sample sizes actually can provide meaningful information that can be used to make inferences about the population as a whole.  According to his "Rule of Five", there is a 93.75% chance that the median of a population is between the smallest and largest values in any random sample of five from that same population.  As it turns out, it's even possible to make inferences about a population from even less data. 

Suppose, for example, that there is a large warehouse filled with large urns containing green or red marbles.  The percentage of green marbles in any one single urn can range between 0% to 100% - in other words, each urn can contain either all red marbles (i.e., 0% green marbles), all green marbles (i.e., 100% green marbles), or a mixture of red and green marbles.  Each urn contains different proportions of green and red marbles - the marbles are randomly mixed in each urn. 

Suppose I randomly select one urn and make a bet on what the majority of the marbles are colored in that urn.  I can choose either green (>50% of the marbles in the urn are green) or red (>50% of the marbles in the urn are red).  If I gave you 2 to 1 odds, would you make a bet with me that I am right?  In other words, if we bet $10 each time I choose an urn (I am not a gambler!), if I am correct in choosing the majority color, I win $10.  However, if I am wrong, you win $20.  The catch is that we have to make the bet with 100 urns.  Statistically speaking, you will stand to win around $500, by the end of the game, right?  I will probably win 50% of the time, and you will win 50% of the time.  Your net winnings are your expected wins (0.5 x $20 x 100, or $1000) minus your expected losses (0.5 x $10 x 100, or $500).  Would you make that bet?  You should!

Now, let's change things up a bit.  Let's play the same game, only this time, I am allowed to randomly select one marble (in such a way that I can't see any of the other marbles) from each urn and look at its color.  Would you still make that bet?  In other words, does the fact that I pick out just one marble from a large urn containing a large, unknown number of marbles that can be any proportion of red or green, provide any additional information to increase my chances of winning and your chances of losing?  It turns out that your bet has become a lot more risky!  Surprised?  I was.

As it turns out, if I randomly select one marble out a large urn filled with marbles, there is a 75% chance that the color of the marble that I select is the majority color for the rest of the marbles in that same urn.  Your estimated net winnings have now changed dramatically in my favor.  In fact, I will win, on average, $2.50 per urn.  In order to understand the mathematics behind this, you have to account for something known as Bayes' Theorem, which describes the probability of an event based on prior knowledge of conditions that are related to that event. 

It seems counterintuitive, but trust me, the mathematics are correct.  We can gain a lot of information from a small sample size, even as small as a sample of one.  What's the lesson for leadership?  Small tests of change can be very, very powerful.  We can make inferences given just a little bit of knowledge - in other words, the old adage that a little bit of knowledge goes a long way, is absolutely true.