Expected Value
A few months back, I came across a concept called “expected value.” It came via way of a poker playing friend, where apparently expected value is a big thing.
I can link you to the Wikipedia article for expected value, but frankly it’s a mathematics-fest and I don’t really understand it.
What Is Expected Value?
(Also, how to know whether you should bet on ANYTHING)
Expected value is a way of exploring options based on the probabilities that they’ll occur.
For instance, if you toss a coin, the expected value of getting a heads is ½ as is the probability of the coin landing on a tails.
That’s a simple equation, but expected value is used for more complicated things. Let’s carry on with the same metaphor.
Let’s say you and a friend are playing a game. He wants to bet you £10 that the coin will land on a heads.
Should you take the bet?
The expected value calculation would be:
- The probabilities of heads and tails are equal.
- If you win, you will have £20. (Your £10 + Your friend’s £10)
- You risk £10 to win £20, and have a 50% chance of winning.
- The Expected Value of this bet is £10.
- Obviously, it’s the same for your friend, as a coin toss is equal.
The expected value of this bet is 0. You risk as much as you’ll win and not get a penny more or less for each bet.
I wouldn’t recommend betting on this, there’s no gain. It doesn’t matter how many times you flip the coin. In terms of probability, you’ll never come out either behind or ahead.
How To Win With Expected Value
My poker playing friend tells me that a lot of poker players use expected value calculations during (and between) their games.
I never really got poker or betting in general. I’m too risk averse (more on that later.) I asked my friend about how he dealt with the risk of gambling, and he told me that with expected value, you aren’t really gambling at all.
You try to only put yourself in situations where you get a positive expected value.
Let’s go back to the coin example.
Imagine that you’ve got a magic coin (bear with me.)You will win 60% of the time.
The calculation looks like:
- You’ll bet £10 to win £20.
- 60% of the time, you’ll gain £10
- 40% of the time, you’ll lose £10.
- 60% * £10 = £6
- 40% * 10 = £4
- £6 – £4 = £2
- Your EV is +£2.
If you bet once, you’ll either have -£10 or +£20. But over time (in terms of probability) you’ll gain +£2 for every bet. The individual bet is irrelevant though. If you played this game 1000 times, chances are you’d come out ahead by around £2000.
So you’d obviously take the bet, should magic coins exist.
What About Expected Value In The Real World?
(Stop Nerding Out Jamie… This Is Worse Than The OODA Article!)
This has all been quite abstract, so let’s use a realistic example.
A person wants to know whether taking a certification course is worth it.
- They currently earn $50,000 a year. That’s average for their field and qualification level.
- The course costs $5000.
- There are hundreds of jobs in their field.
- The average person with the same job that our person has but with the qualification earns $60000.
For working out the expected value, you need to take the above figures and use them something like this:
- If they don’t take the course, it’ll cost them around $10k a year in income.
- If they take the course and do nothing with it, then it’ll cost $5000.
The equation isn’t as simple as “You spend $5k to get $10k a year more – easy money!”
You need to balance up the risk of each.
If there are really hundreds of better paying jobs in the field, then you might have an eighty percent chance of getting a job that’s 10k per year more.
The equation would then be:
80% * +$10k = +$8k
20% * -$5k = -$1k
$8k – $1k = $7k.
So you’d have an expected value of +$7k from taking the course.
Let’s flip it on its head though. What if there weren’t that many jobs, and you only had a 20% chance of getting a better paid job?
20% * $10k = +$2k
80% * -$5k = -$4k
$2k – $4k = -$2k
So you’d have a negative expected value for taking the course.
(Of course, you’d have to take into account that an income raise is a multi-year thing… but this is just an example.)
Why Do I Need To Know This?
If you’re like me and pretty risk averse, then expected value calculations are going to be a major factor in your success as an entrepreneur.
Entrepreneurship is a risky business. It leads the cautious amongst us to sit back and not do anything, for fear we lose time, money or effort.
With expected value, you can calculate risk on a macro-scale, and eradicate the idea that entrepreneurial decisions are “gambling.” This is invaluable.
Final Thoughts
A lot of entrepreneurs talk about “calculated risk.” The problem with that idea is that most entrepreneurs (and people in general) don’t actually make calculated risks. They go on gut instinct and rationalise their decisions afterwards.
That’s why there’s such massive survivorship bias in most entrepreneurial fields. (Copywriting, authorship, e-business… all the stuff I talk about definitely has this.)
Expected value calculations allow you to make real calculated risks, and they allow you to make what appear to be gambles without fear. That’s because if you do the math and only follow positive expected value wagers, then you’ll always come out ahead.
P.S. Sorry for potentially mangling this topic. I’m not sure if this has explained anything. If I’ve raved like a lunatic, let me know in the comments.