0

Now I'm studying elementary probability thoery.

And It changes every abstract notions to strict text.

I mean, when I was in middle school, the probability of two times coin toss is like below. $$P(HH) = 1 / 4,$$ $$P(HT) , P(TH) = 1 / 4$$ $$P(TT) = 1 / 4$$

It is just computed because we have only '4' number of cases and each event of it is just one(1) of the fraction of all events(4)

However, With strict definition, we need to define $σ-algebra$. And If I stand in the time now, I have just only these information : $⌀ , Ω$. So $σ-algebra$ is {$⌀ , Ω$}

And to get the probability, We need to define 'Random Variable' and the 'Distribution Measure'. But, How can I apply this notion to my real case?(I mean two times coin toss)

It is very easy to understand if I just approach this problem with number of case. I mean we have 4 cases, HH and HT and TH and TT So each of it construct the ratio 1 over 4. However when I get the notion : Sigma algebra, Random Variable, Distribution Measure.. with these notion, It is extremly hard to explain that how we derive each probability in the coin toss!

My Question is

Is there any explanation with these abstract tools(Sigma algebra, Random variable, Distribution Measure) to understand above real example (to understand why each coin toss gets the probability 1/4)?

user13232877
  • 101
  • Probability is a choice: there is no answer inherent in any formulation of the situation. The two-coin situation for arbitrary probability distributions for the coin is analyzed from this perspective in the duplicate. – whuber Jun 08 '22 at 16:32
  • The concepts you mention help generalize different notions of probability in order to model much more complex problems for which writing all the possible outcomes and modelling through your approach is practically impossible. Thus, we require abstractions of things such as possible outcomes (space of events), a way to connect them with numbers (random variables) and a way to measure that connection (distributions). For your problem it might not be necessary to use these abstractions, but in order to understand such concepts, these simple problems are very useful to start with. – Jesús A. Piñera Jun 16 '22 at 01:47

0 Answers0