Basic Probability Theory

Basic Probability TheoryThe unfortunate reality is that nothing in this world is certain. In fact, the only thing in life that is certain … is that nothing is certain. This is especially true when we talk about money and investing. Since we can’t deal with certainties, we’re forced to deal with probabilities. Therefore, probabilities must become the lens through which you view all things investment related.

In order to make consistently profitable investment decisions, we have to be able to assess and manage our expectations about risk and reward. We do this by analyzing all relevant information, and then adjusting the probabilities and payouts that we assign to various outcomes.

The framework through which we do this known as probability theory, and a basic understanding of probabilities is important not just for investment purposes, but for life in general. Understanding probability theory, and it’s counterpart – expected value – will go a long way in helping you to become an excellent decision maker.

Why Probabilities Matter

When’s the last time you went to Las Vegas? Or any Casino? Casino’s are the epitome of probability in action.

Every game in a casino is governed by strictly controlled probabilities, and these probabilities always ensure that the house wins … in the long run. This is an important distinction to make: The house doesn’t win every time, but in the long run, across thousands of players, hands, rolls, spins (and drinks, of course) the casino will maintain its edge.

Probability CartoonInvesting works in a similar way. We’re not going to be successful with every single investment we make, but if we can continually keep the odds in our favor, over the long run we’re going to come out on top. Successful investing is not about one or two brilliant decisions, it’s the result of thousands of disciplined decisions over the course of one’s life, all based on rational probability theory.

One of the great aspects of probability theory is that we don’t need to be experts here; a little knowledge will carry us a long way.

The Two Rules that Govern Probability

We’ll begin by going over the two basic principles of probability, and then we’ll play a little game to drive home some key points about probability theory.

There are two properties that govern probability. The first is that a given probability must be between 0 (impossible) and 1 (certain). Or in percentage terms, a probability must fall between 0% and 100%. The second is that all possible outcomes must have probabilities that add up to 1 (or 100%).

Let’s use a basic coin flip as an example to highlight these two rules. When you flip a coin, the probability of landing heads-up is 50%, and the probability of landing tails-up is 50%. Or is it?

Is it possible for a coin to land on its edge? Possibly, but it’s very unlikely. That’s why this result is typically excluded from the possible outcomes. Excluding the possibility of landing on its edge, the only two other outcomes are heads and tails. These outcomes must add to 100%, which is why we assign them equal probabilities (50%).

If there was a legitimate chance of a coin landing on its edge, say 1%, we would have to reduce the probabilities of the coin landing heads or tails. In this case, we could say that the probabilities are: Heads – 49.5%, Tails – 49.5%, Edge – 1%. Here again, all possible outcomes add to 1 (or 100%).

Alright, now that we’ve got that out of the way, we’re going to play a little game to get at the heart of probability theory. And while we’re at it, we’re going to explore a key element of probability theory: Expected Value.

Determining Expected Value

Here’s the question we’re going to discuss: Would you rather have $1,000 right now, or a 60% chance of winning $2,000?

Think carefully about it for a moment. Your answer to this question will involve a number of factors, one of which should be probability theory. This exercise will also help tease out how risk averse you may or may not be.

Keep in mind that there is no “right” answer to this question; your choice reflects how you’ve been ingrained to make decisions. We’ll touch on the behavioral component of making decisions more in a future article, but for now just keep in mind that it’s a good idea to try and understand our own habits and susceptibilities. Do you always shy away from risk, just because? Or are you a true gambler at heart, always seeking the thrill of the unknown? An investor must conquer his or her inner-self in order to achieve success.

So did you come to a conclusion? And did you use probability theory?

The first step in answering this question has to do with creating a fair comparison. We know how much $1000 is worth, but how does that compare against a 60% chance of winning $2,000? To find an answer, we turn to probability theory, or more specifically, a concept called expected value.

In probability theory, the value of a probability based outcome is known as the expected value. The expected value of a particular outcome is equal to the probability of receiving a value, multiplied by the value received. In short hand:

Expected Value = Probability of Receiving Value x Value Received

So, in our case, we can see that the expected value of of a 60% chance of winning $2,000 is $1,200.

Expected Value = 0.60 x $2,000 = $1,200

Why is it called expected value? Because this is the value you could expect to receive if you were to run this scenario multiple times, as in a casino. In reality, the value to you would be either $2,000 (you won) or $0 (you lost).

Now that you have two values to compare for your decision ($1,000 and $1,200), how do you decide? Your individual decision will be influenced by a number of psychological factors, such as risk aversion, which we’ll discuss down the road. For now, we need to highlight something important.

Allowing Probabilities to Work in Your Favor

As previously mentioned, there is no “right” answer to the above question. If you’re risk-averse, you probably chose to keep the $1,000, and if you’re risk-seeking, or have more money than you know what to do with, you might have chosen to gamble for the $2,000. Either decision can be justified.

But if I modify the question slightly, a right answer becomes apparent, and it’s very important for you to understand why. What if I told you we were going to play this game 100 times … now which option would you choose?

The key to allowing probability theory to work in your favor is having enough opportunities for probability theory to take effect. If there is a 60% chance of something occurring, it may not happen on the first instance, or the second, but given enough opportunities, that outcome will occur close to 60% of the time.

If we played this game 100 times, and you chose the 60% chance of winning $2,000 option each time, there is a very high likelihood that you would end up with close to $120,000. This is calculated as follows:

Expected Value = 0.60 x $2,000 x 100 = $120,000

Now, it’s very unlikely that you would end up with exactly $120,000, and that’s okay. The important thing is that we understand the structure of the decision, and know that in the long run this option is MUCH more profitable than choosing to receive the guaranteed $1,000 each time, which would leave you with $100,000 after 100 plays.

Probability theory never plays out over one event, or even a few. When this scenario is played only once, I wouldn’t blame you for taking the $1,000 and running. Some cash is better than none, right?

But if told that the game would be played many times in sequence, if you chose the guaranteed $1,000 each time, I would question your decision making capabilities and assign a lower probability to you achieving long-term financial success. As mentioned earlier, being a successful investor boils down understanding the nature of decisions under uncertainty, and that is governed by basic probability theory.

Exact vs. Subjective Probabilities

In the world of investing, we’re unfortunately never going to be working with exact probabilities. Instead, all we have to go on are subjective probabilities. These are based not on the actual odds of an event occurring, but on the probabilities we assign to an event occurring.

Subjective probabilities are unique to the person making them. This carries two implications. First, recognize that the probabilities you assign to various outcomes will be different than that of others. Second, the accuracy of the probabilities you or anyone else assigns is based entirely on your knowledge and experience. The more you know, the more accurately you will be able to forecast the likelihood that certain outcomes will occur.