The True Function of Probability

After a tumultuous and emotional few weeks in the markets, I’ve finally had a bit of a breather from defending positions. It isn’t over yet but right now, I am moved to write on this particular topic over all the others. Recently, I’ve had the opportunity to engage in debates that involve probability. That is, what is the chance that x event would happen if a particular action were taken? I wrote briefly on this topic near the end of my last post but I just realised that people don’t really understand probability. And here I am touting that the probability of every trade is 50/50 without the mathematical proof! So here is the proof and the understanding of probability that we should’ve all learned in high school but were never taught.

Let’s start off with a stock-related example. Assume that you can only buy shares in a stock or short-sell shares in a stock and that you must choose one or the other. Now what happens after you select either “buy” or “short-sell”? Well, we’d want to know if you win or lose the trade. So the second scenario is whether we win or lose the trade. This is how the choices would look like:

Event Outcome
Buy Win
Buy Loss
Short-Sell Win
Short-Sell Loss

As you can see, there are four possible outcomes with two of the outcomes belonging to each event. The real question is which combinations will result in a “win”? Well, there’s Buy-Win and Short-Sell-Loss. For the first event (Buy-Win), there is a 50% chance for the “buy” leg and a 50% chance of the “win” leg happening. So we can represent this as 1/2 * 1/2. We use multiplication because this is the scenario that results in “true”. For the Short-Sell-Win event, we get the same thing: 1/2 * 1/2. But there are two events and our question can be answered if either event occurs. That is, a “win” will result if we choose “buy-win” OR “short-sell-win”. The “or” is represented by addition. So we can combine the two events by representing it mathematically as:

(1/2 * 1/2) + (1/2 * 1/2)

This results in the following solution:

(1/2 * 1/2) + (1/2 * 1/2) = 1/4 + 1/4 = 2/4 = 1/2 = 0.5 * 100% = 50%

The same is true if the question seeks to find out which combinations will result in a “loss” – it is also a 50% probability. This is, of course, assuming a “fair coin toss” and that no other factors affect the selection or judgment. But as we know, a variety of factors affect our judgment from whether to buy or short-sell to when to buy or short-sell to how much to buy or short-sell. Each decision changes the probability ever so slightly. If this is the case, why then would it still remain at 50% chance of either a win or a loss whenever we place a trade?

Well, let’s think about it for a moment. If you knew that a particular outcome would occur for absolute certainty, then the probability for that outcome is 100% and the probability of occurrence for all other outcomes is 0%. If you had such an instance, I would pour every penny I own into a trade that would result in that guaranteed outcome. But that would mean what? It would mean that I could see the future – I don’t mean by seeing possible futures but to see the actual future that will definitely happen. The reality is that no one can see the future. So although you can have highly sophisticated calculations that show high probabilities of winning trades, at the moment of placing the trade, you are placed in a position of a 50/50 chance of winning or losing. This ratio cannot be changed. Otherwise, it would mean that you can see the future with absolute certainty.

Given all this, why then, would we bother with probabilities? The answer is actually deceptively simple. Let’s take a look at some other examples.

Example 1) If the weatherman announces that there is a 70% chance of a thunderstorm today, what would you do? I would probably pack an umbrella to work. Now what if the weatherman announces that there is a 5% chance of a typhoon today? I would probably leave my umbrella at home.

Example 2) If there is a new debilitating virus with painful symptoms and the only cure is a vaccine that only works 65% of the time, would you take the shot? I probably would and it is very likely that most people would, too! But what if the vaccine is only 10% effective and the virus is not debilitating and has very mild symptoms? Well, I’d probably let my body handle it on its own.

Example 3) The city announces a lottery event where the chance of winning $20,000 is 10% but it would cost $10,000 per ticket. Would you buy the ticket? I definitely wouldn’t! But what if the chance of winning $20,000 is 1% but it would only cost $2 to buy the ticket? Mmmm… I might consider whether I want to give up a Starbucks grande tea that day.

After giving three very generic examples, are you noticing a pattern? If you said, it’s “risk management” then you are correct! Probabilities serve to inform us in our risk management decisions! In each of the examples above, even if you were to make them more sophisticated, to take more factors into account, at the end of the day, it is all the factors and numbers that help you decide whether to take a particular risk or to avoid the particular risk. That’s the chief function of probabilities – to aid us in making decisions! And so when we look at insider traders (the true insider traders – the criminal kind), we see that they’ve stacked the odds in favour of themselves. But is it truly predictable? Of course not! That is why they take various actions to ensure that it’s as close as possible to a “win” situation. For example, they might keep their criminal activity secret and not tell anyone. Or, they might manipulate the market so that it is much more likely to go in their desired direction. But there’s always that unknown – that the Feds might catch wind of their activities or a catastrophic disaster occurs on their planned day and upsets the entire market. It’s never 100%.

So the next time someone mentions to you that the probability of x is y% and that it’s a certainty and therefore, you should take a particular action, just remember that it’s not. They are speaking of experimental probability. The figures could help inform your decisions but at the end of the day, it predicts squat. Probability is not magical.

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