The Mathematical Theory That Could Change Your Life

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Expected_value_continuous_pdf_001.svg Source: Wikimedia

The savviest poker professionals use it to keep a clear head and protect their bankrolls. Business owners use it for effective decision-making. Expected value is a little known mathematical theory and it just might be the thing to help you change how you live your life.

When it comes to making decisions, the most common approach we are taught is to involve our emotions. Even high-level strategists will divide an outcome of a situation into the worst case, best case, and expected case, and then make their choice accordingly. Even a strategy that involves weighing up cost vs. benefits is made from an emotional place rather than considering probable outcomes because it attaches positive associations to the latter.

This is, of course, a perfectly adequate way to strategize and make decisions, but what if there was a solution that took away the emotional weight of a worst-case scenario and instead, gave us the probability of possible outcomes? This is expected value or EV. Its academic definition is the “weighted average of all possible outcomes…the average return that will be made if a decision is repeated again and again”, according to Kaplan. Its formula looks like this:

EV = Σpx

And it is obtained by multiplying the value of possible outcomes (x) by the probability of each outcome (p) and summing the results.

So far, so mathematical! Let’s take a look at a real-world example.

If you were offered $50 to correctly guess which side a flipped coin would land on and $0 dollars for an incorrect guess, you’d have an EV of $25. How? Well, there is a 50% likelihood you would guess correctly and a 50% likelihood you wouldn’t, so the EV calculation would look like this:

(50%)(50)+(50%)(0) f0= 25

If, on the other hand, you would lose $50 for each incorrect guess, your EV would be $0 because:

(50%)(50)+(50)(-50) = 0

Using worst cast/best case scenario, you’d be stuck into thinking that you could lose $50 or win $50 and may not make the decision to play the coin game altogether. However, expected value shows the “worst” probable outcome would, in fact, cost you nothing.

Let’s take a look at another at a simplified example that can be applied to a business situation.

A small business owner with an employee needs to provide a client report. The business owner values their time at $90 per hour and pays their employee $20 per hour. The report would take the business owner 1 hour to complete, but the employee would need twice as long. Using more traditional decision-making methods, the business owner would think it would be better to complete the report by himself/herself because it’s more time efficient. Expected value says something different. They would actually save $50 by delegating the work, even though it would take twice as long because 1 hour x $90 – 2 hours x $20 ($40) = $50.

One of professional poker’s top women’s champions Liv Boeree makes use of expected value in evaluating her lifestyle decisions as well as poker strategy. The life of a pro poker player involves many flights to different countries and continents, which can all add up to spending hours and hours killing time in airports. Liv values her time, so if she arrives for flights too early, she ends up wasting it but is guaranteed to make her flights. On the other hand, if she arrived as late as possible, she’s more likely to experience quicker check-in and security processes, saving her time but may miss the flight. Liv has made a conscious effort to calculate both the negative and positive values of these scenarios, summarizing that missing the flight would only happen 20% of the time: “I think ‘am I going to make this flight if I turn up later, or will I make it 70% of the time? If every minute spent in the airport is a cost to me, how does that balance with the risk of potentially missing a flight 20% of the time?”

By removing any emotional or personal attachments to outcomes and merely looking at them from an equally valuable point of view, Liv and other people who integrate expected value into their lives focus instead on probabilities. This enables the brain to work through typical limitations and see situations with a potential for loss as merely that. Using EV helps you to take calculated risks where you’ve considered the probability and value of each outcome. In spite of the short-term variances, if you can incorporate this strategy into your decision-making, you could find yourself gaining extra time, energy, and finances.


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