Learn, Think & Do

You probably know what does probably mean. A quick google search would reveal that it means almost certainly. So one would expect probability to mean something similar and it is so. But we are not interested in dictionary meaning of it, here we are going to deal with the probability in terms of its mathematical meaning. Don’t worry, I’m not going to use too much of mathematical jargon but some would be inevitable for the proper understanding of it. I would try to keep it as simple as possible. In this blog I would talk about intuitive meaning of probability and in future blogs we would discuss basics of probability, conditional probability, random variables, pdf, cdf, hypothesis testing etc. to set the stage for their use in machine learning.

If you think you are the only one scared of probability or it doesn’t make sense only to you, be assured you are not alone. When I was first introduced to it, I too was confused, not because I did not understand the math behind it but because it made no sense in the real world (let’s not get into philosophical debate about what is real, you get my point). For example, if I roll a die, let’s say a fair die, in reality there is no such thing as a fair die, one would expect each number from 1 to 6 to appear with equal probability, that’s what is called equally likely. The total probability is 1 and each of the number appearing on the die is equally likely, i.e. suppose is the probability of each number appearing on die, so we have . That is, the probability of each number appearing on the die is . Now suppose you roll a die six times, you would expect to get all the six numbers on the die once. Alas! That’s not how probability works. A die is not a computer, it does not have a memory to keep track of how many times a number appears on it, in mathematical terms we say that each throw of the die is independent meaning on each throw each number have again probability of appearing. So probability is useless! Hold on, don’t be disappointed. Here comes the beauty of mathematics into play in the form of Law of large numbers. It basically says in the case of a die that if you throw a die for huge number of times then each number on the die would appear for about th of time i.e. they would get closer to the probability that we have calculated intuitively assuming it is a fair die. Isn’t it amazing! A die which has no memory, behaves randomly for small number of throws, but converges to its intuitive probability given large number of throws. This theorem is basically why probability becomes useful in finance, politics, decision making, ,machine learning and many other fields.

Why study probability?

Before going any further let me give a short answer to why study probability. If you have not been living in wild and have a rough idea about how modern businesses function, you know data is very precious and making sense of the available data is the most sought after skill in today’s world. To do it we use statistical techniques and tools and all these are based, you guessed it right, on the probability theory. For example if you want to understand what it means by there is 60% chance of rain today, or some drug/treatment is effective in 80% cases, or corona antigen test is 95% accurate, or chances of market going up is 70% and so on, you need to understand probability at least in intuitive sense. Later when we go further I would give examples to show that human mind is not well developed to handle uncertainties. So in the modern world if you want to make better, informed decisions then you need to understand basics of statistics which is based on the probability theory.

Some history

The modern axiomatic way the probability is being done was developed by Russian mathematician A.N. Kolmogorov. But initial enthusiasm to probability came from the games of gambling in 17th century over a problem of points which was brought to attention of Blaise Pascal by Chevalier de Méré. It was the correspondence between Blaise Pascal and Pierre de Fermat which led to the solution of the problem and brought probability to the attention of greater mathematical audiences. Later in 18th century Jacob Bernoulli proved a version of Law of large numbers, and Abraham de Moivre postulated a version of another important theorem in probability called central limit theorem. In 19th century probably the greatest mathematician of all time Carl Friedrich Gauss used a form of least squares method to solve a problem in astronomy. Another great mathematician Pierre-Simon Laplace laid down many fundamental results in probability and statistics.

In the next blog I would introduce you to basics of probability.