P&S L1 C5 S4
Intro Stats: Islamic Approach -- Part 2 Prob & Stat
Lecture 1: Random Samples -- Concept 5: Law of Large Numbers
S4: Observed Frequencies can be used to measure probability
The weighing scale does not DEFINE the weight, it MEASURES the weight which is the effect of the gravitational force on the body mass. Similarly probability is INDEPENDENT fo the means used to MEASURE probability. Validity of models does nto depend on measurements, though VALIDATION of models MIGHT well depend on such measurements
The Birthday Problem: What is the chance that in a class of 25 students, two will have a common birthday?
SOLUTION: It is possible to calculate the answer using some difficult combinatoric formulas. However, it is easier to program a computer to find the answer, by making use of the Law of Large Numbers. The first step is to CREATE A MODEL. We assume that date of birth is like a UNIFORM random variable, equally likely to take any value from 1,2,3,...,365 corresponding to the number of days in an year.
REMEMBER: The Model is not the reality -- there are many reasons why the reality may be different from the model. We can do calculations under the assumption that the model is true. It would not be possible to do them otherwise.
Why is this a PARADOX? Intutiive reasoning goes like this. Suppose 24 people have 24 different birthdays. The chance for 1 person to have common birthday with them is 24/365 which is quite small. The chances for such coincidences should be quite small. The CALCULATIONS show something different.
We cannot make any calculations about the complex realities, the actual students in the classroom that is under consideration for probability. We can make calculations within our model, assuming that the model is like reality. Very often, people forget the distinction between the model for reality and reality itself, and this leads to many kinds of confused reasoning.