Mental Model: Expected Value and Variance
Why "Luck" isn't in my vocabulary
What is a mental model?
In Berkshire Hathaway annual meetings, Charlie Munger is known to every now and then analyze a situation using his vast toolkit of mental models.
Mental models are frameworks that assist in one’s ability to make decisions.
I believe mental models can be extracted from any field. Today I will be going over a mental model I use that helps me make better life decisions. Often times, when we are specialized in a specific field, we see the world through a specific lens. Charlie Munger is a polymath who encourages people to study the top 3 textbooks in each field and extract their core ideas. This will allow one to see the world through multiple lenses. I personally have not done this, but I intend to.
The mental model I am sharing today is a model extracted from mathematics and probability. In this post, I will go over how I use “Expected Value” and “Variance” to make decisions.
I first came across the topic of “Expected Value” and “Variance” in the book ‘The Theory of Poker” by David Sklanksy. In the beginning of my poker career, I read this book explaining the mathematical approach and theory behind how to best play poker.
For those who have never played poker before, poker is a card game, and the variant of poker I play is called No-Limit Holdem. In NLH, you get 2 cards that are hidden from your opponents. Each player takes their 2 cards and combines them with 5 community cards to make the best 5 card hand. The player with the best hand wins all the chips. NLH is a game of incomplete information because you don’t know exactly what cards your opponent has. Chess is a game of complete information, where all the pieces are on the chess board, so you know exactly what pieces your opponent has at all times.
You can win in poker by having the best hand or by bluffing your opponent by making it seem like you have the best hand and getting them to fold.
Poker, in my opinion, is a game of skill. To be a winning poker player, you need to be analytical, understand the range of potential outcomes, and understand the mathematics of poker. There are elements of psychology, but being predominantly analytical and understanding the math behind poker are enough to make you a winning player. Poker is a game where rationality and logic reign supreme.
Take a second here to plug my recently launched YouTube Channel: https://www.youtube.com/@continuouscompounding/about
I will be posting stock pitch explanations on my channel. Furthermore, I will be recording my trip to Omaha, Nebraska to attend Berkshire Hathaway’s Annual Meeting. I’ve been wanting to meet my role model, Warren Buffett, and I finally get to do it this year! Make sure to Like and Subscribe!
First, let me explain the concept of “Expected Value”
The easiest example is flipping a coin. There are 2 outcomes: heads and tails. There is a 50% chance of the coin landing on heads or tails. Let’s say I am given fair odds of 1 to 1. Meaning if I bet 100 dollars on heads and the outcome is heads, I win 100. If it lands on tails, I lose 100.
The expected value is the expected payout given all possible outcomes. So in the case of flipping a coin:
Heads: 50% multiplied by (+100) = +50
Tails: 50% multiplied by (-100) = -50
Expected Value = +50 -50 = 0
When given fair odds of 1 to 1, flipping a coin should be a break-even endeavour.
Scenario 1
Let’s say you met Mr. Beast, and he’s feeling charitable. He is drunk and offers you to pick a number between 1 and 10. If a 10-sided die lands on your number, you win 2 million dollars (Option A), or he can simply give you 100k (Option B), which option would you choose?
The expected value of option A is = (1 correct outcome/10 potential outcomes) * 2mm = 200K. Hence, by picking the first option, your expected value is 200K, and you earn an additional 100K compared with option B.
The thing is, people may be risk averse. 100K may mean a lot to them. Think of all the things you could do with 100K. How many of you would pick option A over option B? I think a lot of people, where the marginal utility of 100K relative to their wealth is extremely high, will actually pick option B over option A because there is a 90% chance of getting nothing.
This is where I would like to introduce the concept of “Variance.”
Say you and I play heads or tails 50 times, and I am heads and you are tails. I win 40 times out of 50. Based on expected value, we should both win and lose 25 times. A reasonable outcome is for one of us to deviate from 25 by +/- 3. So 22-28. I’m too lazy to calculate what a one- or two-standard deviation outcome is, but for the purposes of this post, it’s not necessary to go that far.
A key principle you should remember is:
“The dice has no memory”
Or in this case, “The [coin] has no memory.” Meaning each outcome is an independent event. Just because I won 40 out of 50 coin flips does not mean the next outcome must be tails. Variance is a result that deviates from its expected outcome or value. Over enough trials, the results should converge to their expected values, but we never know when they will converge.
Let’s say there is a 1 in 100 chance of getting a $10 McDonald’s gift card from their peelable stickers. You love McDonald’s and eat it almost every day. You have bought 297 McDonald’s and have not won the gift card once. You then win 3 in a row. This is also a potential branch in the decision tree of 1/100 odds.
How do we reduce variance?
A decision must be repeatable for us to reduce variance. If we were able to flip a coin a million times, the final outcome would be extremely close to 50% for heads and tails.
Going back to the Mr. Beast example. Let’s say Mr. Beast sees how much agony you are in and agrees to let you roll the die 10 times at a 200K value each time instead of a 2mm value once. Option B is still the same offer at 100K guaranteed. Now, which option would you choose?
The gross value/total betting volume = 10 dice rolls *200k a pop = 2 mm
Hence, the gross betting value remains the same at 2 mm. Here you have 10 opportunities to make 200K. If you hit your number all 10 times, you make 2mm. If you miss all 10 times, you get 0. If you hit your number even once, you’d make 200k and do much better than picking option B.
The expected outcome of selecting option A is= 10 rolls * [(1/10)*200K]=200K
Which option are you choosing now? A or B? If I were to guess, most of you who chose option B in the first scenario are way more likely to have transitioned to option A in this scenario. The reason is that by allowing you to repeat the outcome 10 times, you have theoretically reduced the variance. Notice how the expected value is the same, but your ability to realize the outcome you want has increased.
How does this apply to life?
Before I get into that, I’d like to go back to poker real quick. Poker is a game of extremely high variance. You can make decisions that have a positive expected value (+EV) but still lose money in the short term. I played a pot once that was worth 3K. I flopped top set, and my opponent shoved all-in with top pair medium kicker. I was a 90% favourite on the flop. My opponent hit two back-to-back cards to make a straight to win the hand. My expected value was 2.7K (90%*3K), but my outcome was 0. As a poker player, you learn to accept variance. You accept that this is part of the game, and since YOU CANNOT CONTROL VARIANCE, you focus on MAXIMIZING YOUR EXPECTED VALUE.
After learning about expected value and variance through poker, I started applying the concept to other aspects of life. Often times in life, you are presented with decisions and opportunities that only present themselves to you once. From a guy’s perspective, let’s say you meet the love of your life. You are 100% certain they are the person for you, but if you screw up the dating stage, she’s lost forever. What do you do? You don’t focus on the potential of losing them because you can’t control that. You focus on making yourself the best possible version of yourself. Here is a list of things you could do in preparation for the first date:
-Shower/Shave (+EV)
-Pick out appropriate clothes for the occasion (+EV)
-Get a new hair cut (+EV)
-Plan out the date beforehand (+EV)
-Wash and clean your car (+EV)
As an oversimplification, any decision you can make that improves your appeal on the first date increases your chances of achieving your expected outcome. Your goal should be to maximize, as much as possible, the expected outcome of achieving a goal. Notice how, in this example, we aren’t being so quantitative. That is the beauty of this mental model. We don’t need to define exactly what our probability of marrying the love of our lives is. We simply focus on things that allow us to increase the likelihood of making this woman our lifelong partner.
How have I, for the most part, wiped “Luck” out of my vocabulary?
Luck is simply variance/deviation from the expected outcome. Most people, when faced with an unfortunate result, linger on how unlucky they were. Typically, people say they are unlucky and don’t look deeper. Lingering on luck is simply lingering on the result. We cannot control the result, so the most productive use of brain power should be on the things we did leading up to the result that may have reduced our expected value.
For example, let’s say you scored 80% on an exam, but your goal was 85%+. I would focus on how much time I had to study for the exam. How many practice questions did I answer? How was my quality of sleep leading up to the exam? At UBC, I found that if I gave myself 2/3 days at 4-5 hours per day to study for an exam, I would almost certainly ace the exam. Whenever I studied for an exam the day before, I let my natural talent for the topic determine my grade. This would often be a hit or miss. But for almost any subject, if I had a full 2/3 days: 1 day to review course material, 1 day to do practice exams, and another day for review, I would maximize the benefits of “spaced repetition.” Sidetracking a little here, I might be butchering the science here, but after a good night’s sleep, your brain’s neural pathways get strengthened, so you process certain things more efficiently and quickly. Sleep is like a save button. If you were to cram 12 hours of study over 3 days vs. 1 day, you are certainly going to perform better over 3 days.
Are you a process- or results-oriented person?
Based on the above study example, I hope you can adopt the framework of identifying variables that are correlated to attaining an ideal result. Once you identify the variables, your goal is to specify in detail how each variable is accomplished. By analyzing each process individually and setting measurable goals, I maximize the likelihood of achieving the expected value. Here is an example of the above in play:
I have a friend I go to the gym with. He recently applied to be a registered massage therapist (RMT). He told me he was worried about getting his certification because you must score 80% or above to pass. I asked him, “How many hours do you think you can study a day?” He said 6 hours. I then ask him, “How many days do you usually study for an exam?” He said, “One.” I said if you gave yourself 2 days to study for an exam, do you think you could score 80% or above? He said potentially. I said, What if I gave you 5 days? He said, “100%.” Notice how I have now primed his mind for the inevitability of passing. I said, Okay, now scale down from 5 five days to the point where you still believe with 100% certainty you will pass an exam. We often confuse talent as the only variable necessary to succeed. There are certainly extremely smart people out there who can simply study the night before and ace the exam. We live in a world where planning, defining the parameters for success, and hard work are underappreciated.
I had a little too much fun my first and second year of college. UBC is on a 4.33 scale. I calculated that in order to graduate with honours, I needed to average 85%+ or score 4.00/4.33 on all my remainder year three and four courses. Here are the specifics of what I did:
-Enter the grade breakdown of each course into Excel. Keep track of my achieved results for assignments, exams, participation, and projects.
-For any segment that had easy marks, I made sure to not take them for granted. i.e. participation marks. I got 9%/10%+ for almost every course.
-I scheduled all my courses after 11am and did not eat lunch. I was not a morning person and found that when I was in a fasted state, my brain absorbed information more efficiently.
-Based on the difficulty of the course, I scheduled the number of days needed to study for each exam. Typically, 2 or 3 would almost certainly allow me to score 85%+ on the final exam.
-The day before the exam, I ensured I got at least 7 hours of sleep.
By getting all the easy marks before the final exam, I found that I often only needed 75-85% on the final exam to achieve an overall course grade of 85%+. Knowing this ahead of time completely reduced pre-exam stress and allowed me to completely focus on studying and performing well on exam day.
By implementing the mental model of EV and Variance, I became a process-oriented person.
Quick blurb on why I believe focusing on the process is more important than focusing on results. Let’s say there are 2 millionaires, Sally and David (fictional example). Sally started her millionaire journey by achieving 1st place in the World Coffee Barista Championship. Utilizing her popularity, she started her first coffee shop, but due to her lack of business experience, she failed. Undeterred, she immersed herself in business books, podcasts, and videos. She also partnered up with an investor who is strong where she is weak. Sally’s next coffee shop is a success, and in 2 years she opened a chain of coffee stores. David is a 9-5 salary man who hates work and wishes every day to win the lottery. His dreams come true, and he wins the lottery. On the day he wins the lottery ticket he raises the middle finger to his boss and all his former colleagues. Let’s say we stripped both millionaires of their riches. Which one would you bet would be able to regain millionaire status? I hope you chose Sally. The skills and experience needed to create a profitable and successful business are not erased. Sally will simply utilize her skills and experience to open up another coffee shop. The lottery ticket winner can only hope his next ticket hits. Based on Sally’s journey to becoming a millionaire, she knows the exact processes that will maximize her chances of attaining her expected net worth of a million dollars.
Being too focused on results and the emotions that come with negative results
As a person who extracts meaning from practically anything, I distinctly remember hearing Poker Pro Phil Galfond say in an interview once that when he is faced with a tough poker decision, he believes it is a waste of time to linger on the emotional side of how a certain poker decision makes him feel, such as the negative emotions that come along with the potential to lose the hand or get bluffed. In online poker, you have a timer that forces you to make a decision in a reasonable amount of time. He says that he focuses on the core factors that matter the most in making the decision. If you have 30 seconds to make a decision and you spend 10 seconds lingering on how the decision makes you feel, you only have 20 seconds to analyze the situation. By not lingering on potential outcomes, he has the full 30 seconds to analyze the hand. In a way, he is maximizing his expected value by getting a full 10 seconds more to analyze the hand.
Now, most of us are not AI robots. We are humans, and we have feelings. When faced with a negative outcome, it is likely you will feel down. Feel sad, recover, and analyze the factors that led to your result. Find areas to improve so that the next opportunity that arises, your EV is higher than it was before. In other words, find comfort in the fact that you have set up your odds of winning to be the highest, and the result will be the result. I have found that by following this mental model, I have become less emotional. The interesting thing is that in poker, I can lose 1K without feeling any negative emotion. I comfort myself by praising myself for making the correct decision. The only time I fault myself is if I didn’t analyze the hand properly. One of the greatest life lessons I have extracted from poker is to focus on maximizing your EV, and the results will come eventually.
What mental models do you use? Comment below; I’d love to engage with you. My readers come from diverse backgrounds, so if you have a mental model you use as well, please share.
If you found this post valuable and believe it might resonate with someone in your circle, please share it. I'm grateful for your support in spreading the word, as it helps me reach new audiences and grow organically. Thank you for being part of this journey!
Follow me on any social media platform you use: https://linktr.ee/continuouscompounding