How to Think About Risk, Probability, and Expected Value

A decision is either good or bad when you make it; what happens after the fact doesn't change the quality of the choice at the time it was made.

About Lucky Maverick

About Jonathan Bales

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Whenever I eat out with my gambling buddies, we play a game called “credit card roulette” to settle the bill. If you’re unfamiliar, “CCR” is a game in which everyone puts in a credit card to be selected at random to pay the entire check. If it’s just a few people, we just mix up the cards, hidden away from view between someone’s hands, then have someone pick a random one to pay (“top card pays,” for example). If it’s a bigger bill with a little more on the line, we might ask the server to get involved, usually selecting one at a time to not pay until just a single card remains. It’s actually quite a fun sweat on bigger bills, so I highly recommend playing if you can withstand the swings.

CCR turns a 100% chance of a small meal payment into a smaller probability of a larger payment. If you’re eating with more than one person, most of the time, you’re going to get a free meal. In a group of 10, you’re eating for free 90% of the time. But, of course, 1-in-10 times you’re paying for everyone to eat.

I’ve run super hot in CCR lifetime. I’d estimate I’ve played 100 times and paid maybe 50% of what I “should have” based on my meals. Of course, that is expected to even out over the long run. The only asymmetry in CCR is if someone continually orders more than you.

I bring up CCR because I think it’s an easy example of how to properly think in terms of probabilities and EV (expected value). Let’s say you go out to a fancy meal with nine friends. The 10 of you live it up with a $2,000 bill and your friend Gary loses CCR and is stuck paying.

Now here’s why CCR is so fun when you win. Not only do you not need to pay a dime for your meal, but you also don’t need to say thanks to Gary. Why? Because you paid just as much as he did.

Thinking about the world in terms of probabilities and EV means not retroactively looking back on decisions as “good” or “bad” or recalculating the math based on one outcome. While Gary paid $2,000, no one knew who was going to pay before playing the game. Yes, it sucks for Gary, but in reality, you took on just as much risk as he did. Your EV was -$200, just like everyone else’s, as you risked a 10% probability of getting stuck with the same check. Gary owes you a thanks just as much as you owe him one; you both paid the same amount.

That might seem ridiculous, but it’s really how you have to think to be a sound probabilistic decision-maker. The fact that you have the benefit of hindsight to see your range of outcomes shrink from a 10% chance of paying to 0% is irrelevant; unless someone is cheating in the game to increase their EV, you should think about the difference between what you paid and what Gary paid as $0, because that is what the difference in EV was when you made the decision to play. You both paid in EV, and it just so happens Gary was unfortunate enough to see his 10% chance of paying $2,000 actually realized—but that doesn’t change the EV.

The EV does change as the game progresses, though, assuming you eliminate one card at a time. This basically functions as a mathematical representation of excitement; the reason it would feel like such a relief to not pay after having your card in the final two, for example, is that your EV immediately drops from -$1,000 (-$2,000 * 0.50) to $0, since once you’re eliminated your new chances of paying are 0%.

A useful tool for thinking about probability is to basically ignore what happened in reality and instead ask yourself what the outcome would be if you could play out the same scenario with the same variables a million times. In that case, you’d see that in your dinner with Gary, you’ll end up paying as much as him—and everyone else—on average: very close to $200.

You can make great strides in decision-making simply by analyzing past choices in terms of their EV and not the results. The outcomes are only important insofar as they affect your next EV calculation.

Probability and Risk in the Real World

In CCR, the only benefit of playing and taking on risk is because it’s fun; over the long run, you’ll pay the exact same amount, but you’ll undoubtedly have more excitement in your life.

In the real world, though, you can change your EV by taking on risk in many situations. Humans are hardwired to avoid risk, and there are strong evolutionary reasons for this. But when it comes to assessing risk in everyday life—decisions that have more to do with your net worth than our survival as a species—it doesn’t make sense to avoid risk.

By making smart decisions, you can basically play a version of real-life CCR in which just by taking on risk—simply in deciding to play the game—you improve your EV. Suppose that during your dinner, half of the people say they don’t want to play CCR and are willing to pay $250 apiece to not be involved. Normally you’d just let anyone who doesn’t want to play out for the EV, but let’s say these friends are so risk-averse they insist on paying a $50 premium to not take on the uncertainty.

That means $1,250 of the $2,000 bill has been paid, leaving five others with $750. Your new EV is (-$750 * 0.20), or -$150. Sure, you’ll get unlucky 1-in-5 times and be stuck with a $750 bill, but just by playing the game, you’re effectively “making” $50 each time.

I believe most of life’s decisions are nothing more than a complex form of credit card roulette. You can choose to not play the game. And sometimes you shouldn’t because the math doesn’t make sense. But generally speaking, it’s +EV to play—to take on the risk. You can capture lots of EV just by playing the game—EV which will eventually be realized if you’re playing it enough.

Even if you’re risk-averse, you’d probably still want to play the game if you could significantly improve your odds. Well, you can. You can work to find edges to increase the probability of success (say, a 5% chance to pay the bill for a 10-person meal), or you can uncover ways to improve the payoff (having the same 10% odds but on a cheaper check).

Yes, real-life wealth creation is nothing more than CCR, except you can work relentlessly and intelligently to shift the math in your favor. You can improve your probability of success. You can enhance the payoff when things go your way. Or, preferably, both.

The only fundamental mistake you can make is not playing the game.

A Note on Fractional Investing

I was pretty disappointed over the weekend to “lose” the auction for this T206 Honus Wagner baseball card.

I went in with a small group of gamblers, for a few different reasons, one of which was just purely that it’s awesome. This is the most iconic sports card of all-time, in my opinion. I can’t feel too much regret because we weren’t going to go up to the final sale price no matter what, but given the scarcity of the card, you can’t help but feel a little let down about missing out.

One of the other people in the deal was Empire Maker—one of the sharpest overall gamblers but probably the biggest degen I know. My only request before we submitted bids was that Empire would not be in charge of the possession of the card. Before the auction, his exact words to me were “I’m probably like 10% to lose it.”

One reason I’m pretty high on premium assets like high-end trading cards and art is because the process of investing in them is becoming democratized via fractional investing. Sites like Rally Rd, Masterworks, and so on have opened up the ability to invest in shares of high-end collectibles. My initial reaction to fractional investing was “who’d want to own a small piece of something?” but then I realized I do this all the time in DFS/investing via equity swaps, for example, and it was exactly what I was planning to do in buying the Honus card, just on a slightly larger scale.

I don’t know exactly how it will play out but I do believe fractional investing is set to boom, and the price of certain assets along with it as the pool of potential investors expands exponentially.

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