How do you think about probability theory?

[ampakine]

I find the probability theory I'm doing in college very difficult until I start wording it all out in my head. If I word it out then there's no confusion about what P(A) represents and what P(B|A) represents etc. but if I don't word it out then I have trouble thinking about it. I think its the notation. To think about it visually I should be seeing Venn diagrams instead of P(A), P(B), P(B|A) etc. How do you deal with this notation in your head? Do you sound it all out, do you visualise the equations as they are, do you convert them into Venn diagrams etc.?

I find the Venn diagram approach just as tricky because I don't exactly know how the Venn diagram of something translates into the real life situation. For example if I want to know the probability that a footballer will score a hat trick I know that in probability theory notation its this P(A∩A∩A) and I can picture that on a Venn diagram but I have trouble seeing how the Venn diagram represents the probability of someone scoring 3 goals in a row. I suppose its all a matter of conditioning my brain so that it automatically knows what the Venn diagram and notation actually represent.

 

[micromass]

Hi ampakine,

I recognize your troubles, for I also had them when I first learned about probability theory. But one day, it all just "clicked" and I could see what I did not see before. Now, I'm just used to the notation. Something similar will happen to you, just keep practising and trying, and one day, it will simply click!

Von Neumann once said: "one cannot understand something in mathematics, one needs to get used to it". This is especially true for probability theory!

After a while, you will start prefering the mathematical notation, since it is precise. While the language is, well..., ambiguous.

Sorry I couldn't give any hints or stuff, but I hope you keep trying!

 

[bpet]

Further to micromass's comments, there are lots of visual tools that are useful to help understand probability, including density functions (histograms), cumulative probability function graphs, quantile functions; for conditional probabilities often tree diagrams are helpful; often geometric methods such as ratios of areas are useful - for the football problem perhaps visualizing a cube in 3d sliced in each of three directions may help in understanding why P(A1&A2&A3)=P(A1)^3.

In general probability theory can be conceptually difficult, which is why there wasn't a solid foundation for the theory until the 1930's. If you're planning to continue your studies it will be helpful to learn a bit of set theory and think of the event space as a set of mutually exclusive possibilities, and probabilities being measures of subsets of the event space.