Understanding Finance Using Probability and Expected Value

To become successful investors we have to think like burglars, because we must be constantly on the lookout for windows of opportunity 😆 So today’s post is about finding those opportunities with the help of math and probability 🙂

In a recent Poll, I asked if readers would prefer to receive a guaranteed $3,000 or an 80% chance of getting $4,000. If you didn’t get a chance to vote, make a choice now, and remember your decision. You might be asked about it later 😉 Here are the poll results (^_^)


85% of you who voted chose the first outcome 😕

*Sigh* (-_-) Folks, this is NOT okay (>_<) Dagnabbit, you guys 😡 It’s time we had a serious discussion about this. I know it’s probably my fault for not blogging about this sooner but today’s concept is uber important and may change the way you look at money forever.

In probability theory, Expected Value means the “expected” average outcome if an event were to run an infinite number of times. And the law of large numbers dictates that the average of the results obtained from a large number of trials should be close to the Expected Value.

For example, a 6-sided die produces one of 6 numbers when rolled, each with equal probability. Thus, the expected value of a single die roll is 3.5.

 \tfrac{1+2+3+4+5+6}{6} = 3.5.

According to the law of large numbers, if we rolled a die a large number of times (like 1,000 times) then the average of the produced values is likely to be very close to 3.5, with the precision increasing the more times it’s rolled.

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Below is a graph showing a series of 1,000 rolls of a single die. As the number of rolls increases, the average of the values of all the results will automagically approach 3.5.

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The same thing would happen with a coin toss. The Expected Value that a coin will land on heads is 50%. So if we flipped a coin a gazillion times eventually the actual data that we witness will come closer and closer to the expected value of 50%.

Now that we understand what expected value and law of large numbers mean we can approach the poll again from a mathematical angle. The expected value of the first outcome is $3,000 as there is a 100% chance of receiving exactly $3,000 every single time. The expected value of the second choice is $3,200 since (80% x $4,000)+(20% x $0)

If math isn’t your strong suit allow me to sum it up for you 😀 Basically we should choose the second outcome every time because it has a higher Expected Value.

So how can we use this information in real life? 😀 Investing in real estate in a foreign province may appear to be risky, but I did it anyway in 2012 because the Expected Value of growth looked better than comparable local properties. And thankfully it has paid off 🙂 because my property in Saskatchewan grew by over 10% while land in B.C. stayed relatively flat. In 2013 I borrowed money at 3.5% against my home’s equity in order to invest in blue chip U.S. companies which historically have an expected annual return of 8% to 10% which gives me a positive spread 🙂 Once again, this decision has paid off very nicely over the last year 😀

Much like the die roll diagram we saw above, my net worth may fluctuate a lot over the short term due to leverage and market volatility. But I make dozens of investment decisions every year, so by the time I’m in my thirties I should have a pretty large nest egg because the annual return from my basket of real estate, stocks, and bonds, has an Expected Value much higher than the current carrying cost of my loans. This is why I can sleep like a baby at night despite having over $500,000 of personal debt 😉

Financial risk can be mitigated because the long-run average of the results of many independent repetitions of an event, like the returns on a balanced portfolio, is made predictable thanks to the law of large numbers. Today’s new investment might fail, and tomorrow’s might fail also. But as long as there’s a solid argument for a positive expected value then sooner or later our ongoing strategy will become profitable 🙂 We can count on the rules of probability because it’s…

14-04-finnmathematical probability

We can’t think about this poll as an isolated event either. In the real world the law of large numbers applies to everything we do and the trials spread across all the financial decisions we make every day, so it’s crucial that we have a consistent strategy. If we go through our entire lives choosing the outcomes with the better “Expected Value” every time we’re faced with such a dilemma, then despite the perceived risks, we will always be better off in the long run than others who choose the safer, guaranteed outcome or someone who flips back and forth between choices 🙂

Only 15% of voters chose the potentially better outcome. As for the rest of you, I’m curious to know if you would change your mind now or continue to stick with your first choice 😀

Random useless fact: The state of finding it difficult to get out of bed in the morning is called dysania

14-04-change-of-gravity probability

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04/04/2014 6:07 am


I don’t think it’s that 85% of your readers misunderstand “expected value”. The problem I had with the 80% of $4K was the 20% risk of nothing. That is far too high a risk for losing out.

Your results would most likely have been opposite if there had been a 20% chance of ($2.8K), 20% (3.0K), 20% ($3.2K), 20% ($3.4K), 20% ($3.6K).

04/04/2014 8:44 am

Great analysis Liquid,

My choice was 100% winning of $3000 without knowing all these calculations : ). Yes, I agree that leverage is the key to wealth, I invest in stocks with borrowed money, but little bit scare to go for big purchases (farm or rental properties) right now.

Alex Yang (@yyangalex)
04/04/2014 8:45 am

i agree with JR. there is a premium that has to be paid (and is probably worth it in many cases) for a guaranteed result. expected value might make sense mathematically, but practically, when the premium paid for the difference in expected value is only 6%, i’ll take the guaranteed $3K any day if you phrase the poll differently to exclude guaranteed result, people may think differently. ie: 80% of 4K vs 50% of 7K i have serious issues with your claim that by understanding expected value, it helps you sleep at night with leverage. You need to understand that just because the long term expected outcome is a certain way, does not mean the short term results are similar. Sure you might have a 60% chance of winning, but if the tail risk outcome is being wiped out, I’d say that is NOT worth it. Markets can stay irrational longer than you can stay solvent! A string of bets gone bad can easily wipe out a mathematically favorable position. This past year has basically been great for anyone. But the most important lesson to be learned from history is that long term investing success is not about how much… Read more »

Alex Yang (@yyangalex)
04/04/2014 8:48 am

always remember law of multiplication by zeros. it doesnt matter what amazing returns you get over the years. all it takes is one multiply by zero to wipe out the whole thing.

04/04/2014 3:03 pm

Agree with commenters above. I suspect the answer would be drastically different if you’d framed it with the law of large numbers in mind.

Eg: You have the choice between two scenarios, which will be run 1000 times. You only get to choose once.

It’s then obvious that Choice 2 is better.

As the question is framed above, if I ran into this choice one day, take it or leave it – I probably would take the $3K. I don’t think that this is outrageously incorrect thinking.

04/04/2014 11:31 pm

Thanks for that article on basic stochastic theory.
Let me ask you two questions though

(1) How do you calculate the probability rates of each investment?
(2) If you don’t do it yourself, how can you guarantee they are correct?

Your way of investment works fine now, but when a series of those 20% probability rates happens, you may run out of money.

The Dividend Ninja
04/05/2014 6:27 pm

Liquid, I disagree with your hypothesis. Actually I think your readers are bang on. They don’t gamble and they don’t like undue risk.. 😉

Putting the amounts aside, because its irrelevant, lets just look at the probability. The probability of a 20% chance of 100% loss, does not outweigh a 100% chance of 0% loss. I would clearly go with a 100% chance of success every time, than take the risk of a 20% loss, and come out with nothing.

The 100% probability with your strategy, is that at some point you will endure a 100% loss.

As an example, I prefer not to roll the dice when I am investing. Hence I buy companies like MCD, KO, TELUS, etc. because they are a 100% guarantee – they are likely to be around a lot longer than me. Of course they are not going to fly to the moon. I do not buy companies like FB, Twitter, or Tesla, because there is always the probability of losing everything. I will clearly forgo the possible gains on these companies for the risk taken.