The infamous coin-flip. It’s been used to start football games and settle disputes about who has to do what for probably as long as there have been football games and coins.1 However, coins provide a couple of interesting ways to explore probability and behavioral biases. We’re going to explore two of these today.
TL;DR Summary
While coin flips simulate a random process, they have some unique characteristics that can teach us some lessons.
Our brains tend to over randomize events which can cause us to mistake a random streak for a real underlying trend.
Certain sequences (even though equally probable) are likely to occur quicker due to the nature of the sequence. If we can “push” instead of “start over” we are already halfway there when we don’t get the result we were looking for.
These characteristics of coin flips can be applied in investing/business in many ways. Three of these are:
Be careful of differentiating between a real trend and a random sequence,
Look for optionality in your investments, and
Try to find situations where your loss is a “push” instead of a “reboot”.
Gambler’s Fallacy and the Martingale Strategy
Two of the most well known aspects of coin flips is that they are random and independent.2 Thus, whether the coin landed on heads (H) or tails (T) last time has no influence on whether it will land on heads or tails this time. The sequences HH, TT, HT, and TH are equally likely (each has a 1 in 4 chance of happening). Extending this concept, we know that (if it is a fair coin and no one is cheating), a sequence of HHHHHHHH (8 heads in a row) is equally likely to see the next flip be H as it is to see the next flip be T. Due to concerns about a fair coin or cheating, we don’t see casinos offering bets on coin flips (outside of the Super Bowl), but if you want to wager on an event that has close to a 50/50 chance, you can choose roulette and bet on black/red or odd/even.3
This led to the creation of the Martingale betting strategy, which is based on doubling your bet each time you lose until you eventually win. Therefore, if I have a 50/50 chance of winning, I start by betting $5. If I win, I collect $5 and start over. If I lose, I double my bet to $10. Now, while I start out down $5, if I win, I get $10 and I’m up $5 ready to start over. If I lose a second time, I’m down $15. No problem, just double my bet to $20. As long as I can keep doubling my bet every time, I’m guaranteed to win $5 (narrator — no, no he isn’t). If you’ve been to a casino and paid attention, you should be able to identify some of the flaws in this strategy pretty quickly. First, most casino games have betting limits. While it may be easy to bet $5 on red, it is probably quite a bit harder to bet $50,000,000 on red. For most of us mere mortals, even having the capital to place a bet of $50,000 strains credibility. So, could the Martingale strategy ever get that far? The answer is yes. If you start at $5, it would take you 14 consecutive losses to get to where your next bet is going to be $81,920 (and at that point, you would have a cumulative investment of $163,835 in an attempt to get $5). The odds of that are really, really, really small (1 in 16,384), but in 2018, Las Vegas saw about 42 million visitors (obviously not all would be playing roulette). Essentially, most of the time you employ the Martingale strategy, you will win. However, the one time you end up losing, you will lose everything you ever won and more (especially since it isn’t really a 50/50 bet).
This leads to a common experiment that I did in classes most semesters (along with many other professors teaching about behavioral issues). A coin-flipping simulation. The class would be split into two groups and one would actually flip a coin 100 times, recording the exact sequence of heads or tails. The other group would imagine that they were flipping a coin and writing down the sequence of heads or tails that plays out in their imagination. At this point, students have covered randomness and know that coin flips won’t come out HTHTHTHT or HHHHTTTT, so they will mix them up a bit. If you have the time, I strongly encourage you to take on the role of the imaginary coin-flipper and write down what seems like a realistic sequence of H and T for 100 observations (it goes pretty quickly). Once you’ve done that, count up the longest consecutive sequence of heads and the longest consecutive sequence of tails. If you are like my students, you probably have a number of something like 5 or 6 (there may be a few with 4 or a few with 7 or 8, but these are relatively rare). Let me generate 20 100-sequence coin flips in Excel. Here are the results:
8, 11, 8, 7, 6, 6, 6, 6, 6, 6, 7, 6, 6, 5, 8, 8, 9, 8, 9, 4 Average = 7 Maximum = 11
Because it is random, the outcomes will change each time, but the average is usually right around 7 and the maximum is typically between 9 and 11. However, every semester the people actually flipping the coin came out with a higher average (usually by somewhere between 1-2 consecutive flips…for example 5.3 imaginary average vs. 7.0 for actual flippers).
Why does this occur? The answer is our brains don’t do a good job of simulating randomness and we tend to over-randomize events. We are perfectly good with 2 or 3 heads in a row, but once we get to 4 or 5, the tail is “due” in our minds. Thus, if we see something happen 3 times in a row, we are comfortable arguing that it is just random. However, if it happens 6 times in a row, that is too much. However, let’s look at the math. The probability of flipping 6 heads in a row is 1 in 64. That should be pretty rare. However, in our Excel simulation, we saw only 2 of the 20 trials where we did NOT have a sequence of 6 or more and we saw 7 or higher in 10 of the trials (despite it only occurring 1 in 128 times). Why is this? The answer is two-fold. First, there is a 1 in 128 chance of 7 heads in a row, but also a 1 in 128 chance of 7 tails in a row. We didn’t specify that we were looking for heads or tails…just the longest sequence. Also, the odds of 7 heads in a row is really rare if you are only flipping the coin 7 times. However, you had a chance of 7 heads in a row on the first 7 flips, on flips 2-8, on flips 3-9, etc. So you actually had 93 chances for it to occur (or 93 chances for 7 tails in a row).
HH vs HT and Markov Chains
Hat tip to Twitter contributor 10K Diver for this one.4 Consider two sequences of coin flips — HH vs. HT. Let’s say that Alice wins if HT comes up first and Bob wins if HH comes up first. If we play this game 100 times, who will win more frequently? While each is equally likely…if you remember probability, (and we have fair, independent flips) getting HT has a 0.25 probability (0.5*0.5) and HH also has a 0.25 probability (0.5*0.5). However, it is a bit trickier than it appears at first glance (yes, I initially fell into the trap). Let’s look at what has to happen for Alice to win
And then for Bob
Notice that when Alice or Bob start out with a tail, they both lose and must restart. If they start with a head, they both go on to flip two. Therefore, they start out on equal footing. However, in flip two, Alice will win with a tail…but not really lose with a head (as that is what she needs on her first flip anyway). Therefore, it’s more of a push than a loss. On the other hand, if Bob flips a heads on his second flip, he wins. However, if he flips a tail, it is a loss and he must restart from scratch. Therefore (and I apologize for the math for those with a bit of a math phobia — I get it…feel free to skip the section below):
The average number of flips for Alice (4) is calculated as follows:
Let X = the expected number of tosses and
Let Y = the expected number of tosses if you have just thrown a H
X = (1/2)(X+1) + (1/2)(Y+1)
Y = (1/2) + (1/2)(Y+1) ==> Y = 1/2 + 1/2Y + 1/2 ==> 0.5Y = 1 ==> Y = 2
Then, substitute Y =2 back into the X equation
X = 0.5X + 0.5 + 0.5(3) ==> 0.5X = 2 ==> X = 4
Thus, it will take an average of 4 flips for Alice to get her sequence of HT.
Now, let’s do Bob (to get his 6 flip average for HH)
Let X = the expected number of tosses and
If you start with a T, you have wasted 1 toss and must restart (there is a 0.5 probability of this occurring and it will lead to X+1 tosses)
If you start with a H and then get a T, you have wasted 2 tosses and must restart (there is a 0.25 probability of this occurring and it will lead to X+2 tosses)
If you start with a H and then get a second H, you win (there is a 0.25 probability of this occurring and it takes 2 tosses
X = (0.5)(X+1) + (0.25)(X+2) + (0.25)(2) ==> X = 0.5X + 0.5 + 0.25X + 0.5 + 0.25 ==> X = 0.75X + 1.25 ==> 0.25X = 1.25 ==> X = 6
Thus, it will take an average of 6 flips for Bob to get his sequence of HH.
In other words, even though the chances of HH or HT are exactly equal on any 2-flip sequence, we can expect Alice to see her HT (on average) quicker than Bob will see his HH. This means Alice will win a little more frequently.
Real Life Applications of “Coin Flips”
How does this translate to business and finance? The answer is in a multitude of ways.
Seeing “Trends” Which Don’t Exist: This is one of the harder aspects with randomness. There are REAL reasons why one company performs (better management, competitive advantages, optionality, etc.) and, while many of these may be priced in, some may not be. However, there is also a high degree of just random fluctuations. As our brains are wired to see narrative (aka narrative bias),
we may start to concoct a story when we see three quarters of outperformance —even if there is no story there. Consider watching the news when the Dow Jones Industrial Average rises 200+ points and the talking head tells you why. In order to calculate the DJIA, the value of 30 stocks are added together and divided by 0.147 (approximately). Since the DJIA is currently around 35,000, a 200-point move is less than 0.6%. How likely is it that there is one explanation for 30 stocks in a variety of industries moving by less than a percent? However, this narrative bias leads to increased interest in trends, buying “winners”, etc. Differentiating between random noise and real trends is hard.
Companies Should Emphasize Optionality: Optionality just means that if something is a success, it can lead to another opportunity. Consider the Marvel Cinematic Universe which has made Disney and its shareholders something like $1,000,000,000,000,000 (okay, a slight exaggeration…but its made a LOT of money). One of the benefits is that success of one movie leads to the success of the next. Guardians of the Galaxy II was a result of Guardians of the Galaxy. If the first one would have bombed, Disney could have made major changes or just stopped and spent money elsewhere. Now Disney has extended that to Disney+, its streaming service. Not being forced to “start over” each time is incredibly valuable. Even a “one-product” company like WD-40 actually has many products (such as 2000 Flushes) as it allows them to take advantage of their logistics expertise. Look for investments where the brand or process can be extended to new opportunities.
Try to Find “Push/Win” Situations Over “Lose/Win” Situations: Starting over is the killer in business and investing. If you are in situations where the bad outcome is a “push” instead of a “loss”, you are creating a significant benefit. This is one of the arguments for diversification. If you spread your investments across 20+ investments, it is really hard for any one (or even any two) to kill you. Alternatively, if you have a concentrated portfolio with five positions, any one of them can do significant damage. And if two get hit…YIKES! This isn’t to say that a 5-stock portfolio can’t do better. In reality, the concentrated portfolio has a greater chance to outperform AND to underperform (due to the large number of stocks in the less concentrated portfolio benefiting from less volatility). It’s just that the times it does underperform, it creates much more damage. Too little diversification, too much leverage, anything with a high risk of pushing us to zero (or even taking a large loss of 70%+) takes a long time to build back from. Remember that when we have a 70% loss, we need a 233% return to get back to even. You can even apply this to personal decisions — if the action you are taking leads to a dead end, forcing you to start over, it is a worse result than one which allows you to branch into a different direction. That doesn’t mean don’t risk it, but make sure you recognize the risk.
Coin flips apparently date back to Julius Caesar.
Well, for the most part. There is some work that suggests the starting point can influence the result (flipping a coin heads up will create a small edge of about 1% that it lands on heads).
Note that these are not truly 50/50 bets as American roulette wheels have two green “house” slots of “0” and “00” to create 38 potential outcomes (European wheels have one “house” slot of “0”), so betting on red (black, even, or odd) actually provides a 18/38 chance (47.4%) of winning and a 20/38 chance (52.6%) of losing.
If you are on Twitter, he’s definitely worth a follow…just be sure to “grab a cup of coffee”.