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Many students of probability theory (well, this author at the very least) have stumbled when they are confronted with the birthday paradox:
What is the smallest number of people that need to be in a room for there to be a greater than 50% chance that two people in the room share the same birthday?
At first glance, it seems like you have to cram a lot of people into the room, at least 150, possibly 180, or maybe even 200. However, with the assumption of 365 equally likely birthdays (mathematicians have no use for people born on February 29th), the correct answer is a measly 23. As a quick check on this result, there are 23 employees in the EWM main office in Brighton; and, indeed, two people do share a birthday, on April 28th. You can find the mathematical derivation of the result in many places on the internet, but it is important to not get lost in the algebraic manipulations of the solution; the salient point is why this is called a "paradox"? Why are so many people's intuitions about the problem incorrect?
There is a cognitive bias at play here that pops up not just in pretend probability problems, but also in personal finance and in broader phenomena throughout our world. To better understand where my own intuition went wrong, this author imagined a scenario where there is a room initially only occupied by himself. People now enter the room one by one asking all the current occupants "When's your birthday?" to see if there is a possible match. Look at the chart below:
Now without the help of a visual representation of all interactions, I focused on each new person as if they were just the same as the previous people; it seemed to me that the chance of a birthday match should only increase linearly as someone new enters the room. However, as you can see in the chart, with each new person, the increase in the number of possible birthday matches gets larger. The sum total number of possible matches is growing exponentially. My cognitive bias treated every additional person like we were the only two people in the room just like in the beginning of the scenario, ignoring the surrounding accumulating number of possible birthday matches.
The late physicist Albert Bartlett delivered several famous lectures on humanity's inability to comprehend this idea of exponential growth, including many financial examples. For instance, he mentioned one experiment where skiers in Vail, Colorado, were willing to accept a 7% yearly increase in their ski passes, but they were shocked when they were later informed that this meant that the price would double in ten years (the math equation: (1.07)10 ≅ 2 ). They had focused on the seemingly small initial increase, thinking that only this would continue through time; unable to fully grasp how much increases would compound on top of increases through the years.
The standard examples used to illustrate the importance of exponential forces in finance are the money accrued in a savings account or the debt accumulating on an unpaid credit card, but the principle and our difficulty in grasping it are more far-reaching. If the S&P 500 is up 10% for a given year, the value we extract from this fact depends on and is compounded by our earlier financial decisions. Did we consistently add to our portfolio in the past? Or did we just recently start investing in the market? The ability to recognize that consistent positive (or negative) actions don't just have the small easily visible present effect, but build and compound to better (or worse) results in the future may truly be the most powerful force in the universe.