I was wondering, why we need to study Probability. I have come to the following personal conclusion:
A lot of things take place in our everyday life, or we get involved in a lot of things where we can not predict anything for certain beforehand, but we can conclude some forecasts. For instance, we can not answer the following questions for certain, but we can give some forecasts:
- What would be the share price like for the company 'X' in the next month?
- How fast or how big the economy of a country 'Y' would grow in the next quarter?
- How likely is a region to be hit by earthquake next month?
- … … …
Again, there are some problems in our life where the incident depends on pure luck, but, we have to come to a definite conclusion. For instance, we need to answer the following question in out daily lives:
- Who would be declared a winner if a Cricket match is abandoned half way through?
- How would the batting target be like If a Cricket match is interrupted by rain and wasted almost half an hour in the process?
- How would two players share the bet-money if a gambling match is abandoned half way through?
- … …
Now, the question arises, why isn't inferencial statistics good enough to answer these questions?
Well, the answer is, in these cases, either we don't have sufficient data in our hands to be analyzed (e.g. gambling match), or, analyzing a large data doesn't help much (e.g. even though we may have a large amount of data for a company to be analyzed, a share price can drop or rise at any time for arbitrary reasons).
Is my understanding correct?
If my understanding is correct, why do we analyze weather data using probability and Random Processes, as weather forecasts do not change much on yearly basis?
Best Answer
The simple answer is that Inferential Statistics simply cannot exist without probability.
Every major result in Inferential Statistics has a rigorous underpinning in Probability/measure theory.
The Laws of Large Numbers say obvious things: "The sample mean will converge in probability/almost surely to the true population mean", but how on earth would you prove this without formal probability axioms?
But then go further, what about the completely counter intuitive results like the central limit theorem. Why on earth should we expect the sample mean and sample variance to be asymptotically normal?
Finally, consider Bayesian statistics. How could you possibly undertake Bayesian inference without a proper understanding of conditional probability?
On the surface inferential statistics may seem to be common sense results. However, a good chunk of those common sense results require extensive and rigorous proofs, which is where probability theory is important. Without probability theory, statistical inference would not have any of the important results used every day.