Thought process improvement on Probabilities

[cdux]

I had a problem that said "I throw nine balls randomly in five containers. What's the probability any container will have exactly five balls by the end of the process?"

The answer involved using a Binomial distribution.

Now the question is, how do I train my mind to go to the use of distributions and not basic Kolmogorov probability functions? I was spending so much time trying to find a simple answer and I was surprised it was easier with a binomial distribution. What is the right thought process to easily direct me to the use of a distribution in that problem?

 

[mathman]

Use a multinomial distribution.

http://en.wikipedia.org/wiki/Multinomial_distribution

 

[D H]

Now the question is, how do I train my mind to go to the use of distributions and not basic Kolmogorov probability functions?

The same way you trained your mind *not* to use the epsilon-delta definition of the derivative.

That limit concept is absolutely essential to making the concept of the derivative rigorous. That doesn't mean you need to use it to compute (for example) the derivative of exp(-x²/2). It's too much work.

The Kolmogorov axioms are similarly essential to making the concept of probability rigorous. That does not mean you should go to first principles and use those axioms to solve every probability problem.

 

[cdux]

Yes, but why should you not go there? I deal with a list of questions that may use basic principles or they may not use basic principles. I'm trying to find out why on that particular question for example I couldn't automatically think of using a distribution.

 

[blue_raver22]

As a scientist, it is important to have a strong understanding of probability and its various applications. One way to improve your thought process when faced with probability problems is to familiarize yourself with different probability distributions, such as the Binomial distribution in this case.

The Binomial distribution is often used in situations where there are a fixed number of trials and each trial has a binary outcome (e.g. success or failure). In the problem given, we have nine balls (fixed number of trials) and each ball can either be placed in a container (success) or not (failure). This makes the Binomial distribution a suitable choice for solving the problem.

To train your mind to think about using distributions, it is helpful to practice with different types of problems and familiarize yourself with which distributions are commonly used in each scenario. This can be done through studying and solving various probability problems, as well as practicing with real-world data sets.

It is also important to understand the assumptions and limitations of each distribution, so that you can determine when it is appropriate to use them. For example, the Binomial distribution assumes that each trial is independent and the probability of success remains constant throughout all trials. If these assumptions do not hold true in a problem, then another distribution may be more suitable.

In summary, to improve your thought process when it comes to probabilities, it is important to have a strong understanding of different probability distributions and their applications. With practice and familiarity, you will be able to easily identify which distribution is most appropriate for a given problem.