Introduction to Probability Theory, Part 1

In this series of tutorial-style articles I recap what I have learned about probability theory from studying the work of E. T. Jaynes (available online here (book) and here (lectures)), which I recommend - with some reservations. The introductory parts are easy to read and enjoy. However, the later chapters are dominated by references to physics and mathematical formulae whose explanations are rather too brief for my taste. Jaynes seemed to write for students of physics at graduate level (even though I believe it was not his intention). I feel that his ideas are so intriguing and general that they deserve a broader audience. The goal of these posts is to introduce the most important concepts with fewer assumptions about the reader's level of mathematical sophistication; and to verify my own understanding in the process.

It's not just about coins and dice!

If you are like most people, you were introduced to the concept of probability at school with examples such as throwing dice, flipping coins, selecting cards from a deck, spinning lottery wheels, pulling colored balls from urns and other such. You will find plenty of such examples in various tutorials on the web, too. While there is nothing wrong about them in general, they can leave the impression that this is what "probability theory" is all about. A rather boring application of basic arithmetics to some idealized useless "random experiments" that noone cares about in real life. That is, unless they are after good marks for mechanical answers to silly questions like "what is the probability of scoring more than 2 but fewer than 8 with two dice". It appears just about as exciting and thought-provoking as solving quadratic equations for sports.

What they usually don't tell you is that probability theory describes what you - and everyone else - have been doing for your whole life with more or less success, without even realizing. All kinds of reasoning and decision making depend on probabilities that people assign to various propositions:

  • Whenever you look at something (like Escher's drawing of waterfall on the left), you unconsciously figure out the probabilities of seeing different scenes. You make up your mind what the scene is about and whether it is "real" or not;
  • Before you cross a street, you unconsciously figure out the probability of being hit by a car and getting to the other side safely;
  • Whenever you decide to buy something, you figure out the probability of getting good value for your money;
  • Detectives figure out who dun it based on probabilities of finding particular criminal evidence;
  • Criminals figure out how to reduce the probability of getting caught;
  • Scientists figure out which explanation is more probable than others for an observed phenomenon;
  • Businessmen figure out which deals are more likely to bring them profits;
  • Politicians figure out which public statements are more likely to bring them voters;
  • and so on, and so forth.

The really important thing to notice here is that we are almost never 100% certain about anything. We can be rather sure or rather doubtful about different things, but we can hardly ever honestly proclaim: "I know it's a sure thing" or "I know it's completely impossible" - except perhaps when trivial and uninteresting stuff is concerned. To put it in a slightly different way, whenever we need to think and make choices, there is always some uncertainty involved.

Real applied probability theory is about systematically improving our everyday thinking and decisions:

  • It's about drawing the best conclusions from whatever we already know and understand;
  • It's about not getting fooled and confused;
  • It's also about knowing how to act to become more knowledgeable about stuff that matters.

The concept of probability is quite difficult to grasp, though mathematically very simple. A tiny little part of it is about throwing dice and shaking urns in the classroom.

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