Forecasting
In the Unlikely Event
February 23, 2024
Improbable phenomena occur every day. People travel to distant countries and see friends from their hometowns. Demonstrably worse athletes beat their opponents. People select the correct card in games of Three Card Monte, a scam in which the location of the correct card is purposely obscured.
Outside my office window, for instance, a large tree rests on its side precisely where it fell on a neighbor’s property during a recent storm. Happily, although improbably, the tree fell in such a way that it completely missed a nearby house (along with its inhabitants), the street, and shockingly, the electrical lines. Sadly, there has been no progress in the removal of said downed tree, but I digress…
Quantifying the probabilities of such real-world events is challenging. Instead, let’s think of a more constrained problem: tossing a fair coin and getting 10 heads in a row. Or, to use the more arcane and esoteric language of numismatics – 10 consecutive obverses. On its face (so to speak), this event would seem to be vanishingly improbable.
Coin tosses (for fair coins) are statistically independent events, which means that the outcome of one toss is unrelated to the outcome of any other toss. No matter how many Heads you toss consecutively, the next toss will still have a 50% chance of turning up Heads. The probably of 10 Heads in a row is thus the product of each probability:
Let’s consider that there are plenty of other less likely occurrences:
- Being struck by lightning during your life: 1 in 15,300
- Being killed in an airplane crash: 1 in 11,000,000
Both examples are tragic and horrific. However, you are far more likely to sit at your kitchen table, toss a coin 10 times, and get 10 Heads in a row, than you are to have either of these events befall you.
Now, let’s try to simulate the coin-tossing results in Excel (see an example), where we can use the RAND function, which “returns an evenly distributed random real number greater than or equal to 0 and less than 1.” The following equation returns 1 (i.e., TRUE) if RAND returns a value less than or equal to 0.5, indicating a 50% probability. In this case, we are assigning an outcome of Heads to values less than or equal to 0.5. In each of the cells labeled Trial1, Trial2, … Trial10 (in the image below), the following formula appears:
=(RAND()<0.5)*1
Person 1, 2, … 1023, 1024 represent individuals tossing a coin 10 times. The columns labeled Trial 1, Trial 2, … Trial 10 represent the 10 coin-tosses, where 1 indicates Heads and 0 indicates Tails. The Count column indicates the total number of Heads for each person:
Person 1 has 5 Heads.
Person 2 has 6 Heads.
…
Person 1023 has 5 Heads.
Person 1024 has 7 Heads.
...
The following figure displays the Count column for each of the 1024 simulated people (in order from Person 1 to Person 1024), corresponding to the number of Heads tossed by each person, with the single value of 10 highlighted indicating the individual who tossed 10 Heads in a row.
The Excel RAND function recalculates each time the spreadsheet is updated. Consequently, we can view multiple sets of results.
In the following, two people tossed 10 Heads in a row. As a related matter, two people tossed 0 heads (i.e., 10 Tails) in a row.
In the following, one person tossed 10 Heads in a row and one person tossed 0 Heads (i.e., 10 Tails).
If we run through this process many times, the value of 10-Heads will converge to 1 in 1024. The Law of Large Numbers describes this phenomenon.
Before having worked through the math, I would have guessed that you would need more than 1024 trials to get 10 Heads in a row. Here is the takeaway: improbable events will occur if we run enough trials. Unfortunately, we cannot re-run the universe to change the factors that contributed to the tree’s demise to see how it might have fallen differently.
In the context of energy forecasting, unlikely (yet consequential) events include extreme weather conditions and dramatic changes in economic conditions, each of which can be evaluated for their probability of occurrence.
