L2 C5 S1

S1: Passage of Time in Model

It is very important to look at the role of time in applications of probability. This is never done satisfactorily in conventional textbooks. First we start with the simpler situation, and deal with the passage of time in our MODEL of probability, and not in the real world.

MODEL: Let us go back to a Population P with N elements labeled 1,2,...,N and consider a simple random draw R from this population. The random draw R has an equal chance of being any one of the members of the population. We can think of the random draw as an EXPERIMENT which is being performed on the population. We set up some mechanism or procedure to ensure that all members get an equal chance of being chosen, and then we CARRY OUT the experiment. When it is done, then some ONE member of the populations gets chosen: suppose it is person K. Before the experiment, every one has an equal chance of being chosen. AFTER the experiment, all of these chances have disappeared -- one person has been chosen and all other possibilities have been DESTROYED -- it is NO LONGER possible for them to be chosen. There are no probabilities any more -- the choice that has been made is 100% member K and 0% possibility for the other members. So the situation changes drastically BEFORE and AFTER the experiment. Without clarity about this distinction, it is very difficult to get an understanding of how probability works. Accordingly, we set up some notation and terminology designed to capture this difference, which occurs with the passage of time, in our model.

We will use boldface R as a symbol for the random draw. The random draw EXISTS only before the experiment. After the experiment is carried out, there is no randomness left. The probabilities and uncertainties and potential choices all exist in the pre-experimental world, but they DO NOT EXIST in the post-experimental world. Once the draw is made, a particular OUTCOME occurs -- let us call this outcome R, so the letter shows that this outcome is what happened when the experiment R was carried out. This outcome is not known and does not exist in the pre-experimental world. In fact, R and R cannot mutually co-exist. Existence of one drives out the other. We can say that with the passage of time R turns into R. The random draw PRODUCES an outcome. We will use a special notation to show this relationship between the experiment and its outcome: We will write that R => R The equal sign is followed by the arrow of time to show that with the passage of time the random draw R produced the outcome R.

As we can see -- the outcome ONLY exists in the post experimental world, while the random draw, probabilities, possibilities, all exist ONLY in the pre-experimental world. Keeping clear on which world is being talked about is CRUCIAL and CENTRAL to understanding probability. Failure to distinguish the two is the source of MANY confusions, in students, teachers and textbooks.