Question

Design your own experiment to demonstrate the probability of states using objects of your choice. Use Probability of States as a guide for experimental design.

Answer 1

Suppose I am interested in finding the probability that, it will rain tomorrow, given it rained today, let's say for, summer season.

More specifically, I am interested in knowing the four conditional probabilities of raining/not raining tomorrow given it rained/not rained today. Hence, we have two possible states: rain/ no rain (denoted by R/N) respectively. Thus, we get a 4x4 transition matrix out of them:

Now, let whether it rains or not in the nth day is given by a markov chain:

where Xn takes values in {R,N}.

Let's do an observational study. Let's collect the data over year's summer on whether it rained on a particular day or not. Thus, we have data as observed value of the markov chain.

Consider the doublet counts:

Then we can estimate:

[This is a common sense estimate: We are just taking the proportion of days where it rained on the consequtive day as well, among the days that rained.]

Similarly other transition probabilities can also be calculated.

Hence, in this example we demonstrated an experiment involving probability of states. A similar model can also be used to find probabilities like, that of stock prices going up on a subsequent day given it's up in the current day, etc.