Do you see what I did with the title there? Anyway, you must have heard the term ‘probability’ being used around you. People use it in different contexts and in different forms – “What is the probability that Spain will win the next world cup?” or “I will probably finish reading the book by midnight” or “It’s quite probable that she won’t return until tomorrow”. When people talk about probability as a mathematical concept, all they think of is the percentage chance of something happening. But is that all there is to it? If that is the case, then why did they have to dedicate an entire branch of study to this? Probability theory is much more than just calculating the likeliness of something happening. It’s used almost everywhere, by almost everyone, for almost everything. Surprised? Well let’s find out then.
Am I missing something? What’s the big deal here?
Let’s start with an example by considering Michael and Walter. Michael has no idea about probability theory and doesn’t care much about it either. Walter, on the other hand, is a mathematician. Here’s the conversation:
Walter: I have two closed boxes here. The first box has 11 red balls and 3 white balls. The second box has 4 red balls and 12 white balls.
Michael: Alright. So what?
Walter: All the balls are exactly identical so that you cannot differentiate by touching them.
Michael: I hope you are going somewhere with this story!
Walter: Can you pick out a red ball for me?
Michael: How can I do that? The boxes are closed.
Walter: Yeah, that’s true. I want you to take a guess. Just choose a box, put your hand in and pick out a ball.
Michael: What difference does it make? Since I don’t know which one is going to come out, I can just randomly choose a box and put my hand in.
Walter: Are you sure that it will not make any difference if you choose either box?
Michael: I am not clairvoyant. So yeah, I don’t think it’s going to make a difference. It’s going to be totally random.
Do you see the problem here? The first box has a total of 14 balls, 11 of which are red. The percentage chance of getting a red ball if you put your hand inside the first box is 78.5% (11/14 = 0.785). The second box has a total of 16 balls, 4 of which are red. The percentage chance of getting a red ball here is just 25% (4/16 = 0.25). If Michael knew even a little bit about probability, he would have made a better decision.
This is actually a very simple example. Many processes around us are probabilistic. We are just not aware of it. The technologies try to maximize the probabilities of events happening so that the required goals can be achieved. This includes our phones, wireless internet, microwave, TV, manufacturing chipsets, materials used for construction, designing structures and many more. All these things are built upon a very strong foundation of probability theory!
If it can’t predict the exact future, then why do we need it?
We definitely cannot predict the exact future. But that doesn’t mean we have to be blatantly stupid about making our decisions. We can certainly maximize our chances by making the right choices. A lot of times, something called ‘confidence measure’ is used to describe these events. When we see the weather report, we can see that they predict rains with a certain confidence and it’s never 100%. If it’s 100%, then it will be like predicting the future (which is not exactly possible by using the current technology). If they say it’s going to rain with 95% confidence, there’s a pretty good chance that it’s going to rain.
If a sports team has lost it’s previous 10 games and you are asked to predict what’s going to happen in the next game, what would you say? You might say it will depend on the opposition. What if you don’t know the opposition? You might say that depending on how they have been playing in the last 10 games, they will probably lose. Given a bunch of observations, you formulated a model and predicted something. It’s not impossible for them to win this match, just that it’s very unlikely to happen.
What is this ‘Gaussian’ thing I keep hearing about?
Given a bunch of observations, you can say that something is going to happen with certain confidence. Just like you did earlier with the sports team! You definitely cannot predict with absolute certainty. Sometimes you are more confident and sometimes you are not so confident. This concept is consolidated in mathematics and is given the name Gaussian distribution. This is a model to analyze and predict the chances that some event is going to happen. That something is equivalent to ‘mean’ of the distribution and the ‘confidence’ is equivalent to variance of the distribution. If a curve has a sharp peak, it means that the event is very likely to happen. The concept of Gaussian functions and distributions is probably one of the most important concepts known to mankind. It appears so frequently in so many different fields in so many different forms that it wouldn’t be wrong to call it a fundamental building block!
I can write another few thousand words about the mathematical concepts of probability theory and its applications in real life, but I will reserve it for another blog post. Depending on the number of interesting things I have to say about this topic, I would say we are looking at a trilogy here!