The Irrationality of Probability
“I believe that we do not know anything for certain, but everything probably.”
― Christiaan Huygens
We would all be familiar with the concept of probability, or at least would like to think so. Defined as the measure of the likelihood that an event occurs, most of us believe it to be the ratio of the number of favorable outcomes of an event to the number of total possible outcomes.
P(E) = (Number of Favorable Outcomes)/(Total Number of Possible Outcomes)
This is what the Classical Theory of Probability states, agreed. But this assumes that every outcome has an equal likelihood of occurring. It may very well be valid in case of a set of fair coins or a set of fair die, but just conducting a simple experiment with a combination of the two would yield a problem that cannot be solved using this approach. Consider the experiment of tossing a coin wherein if the coin shows Heads, it is tossed again, but if the coin shows Tails, then a dice is thrown. The probability distribution is like this:
It can clearly be seen that the probabilities are not equally distributed and cannot be solved using the previously mentioned definition. Now, that was just one drawback of the Classical Theory of Probability. Another very important drawback is its failure to address irrational probabilities; after all, how many times have you heard of the probability of an event to be √2 or π or log 5? Rarely ever. So how do we define probability such that these shortcomings could be addressed?
Using the Axiomatic Approach to Probability first developed by A. N. Kolmogorov, it is defined as a function P with the following axioms:
i) P(S)=1, where S is the sample space of the experiment
ii) P(E)≥0, where E is any subset of sample space of the experiment
iii) P(A⋃B)=P(A)+P(B), given that A∩B=ϕ i.e. A and B are mutually exclusive events
Using this modified definition enables us to find conditional probabilities of events and determine probabilities with irrational values, for instance, the Buffon-Laplace Needle Problem.
Buffon-Laplace Needle Problem: Find the probability P (l; a,b) that a needle of length l will land on at least one line, given a floor with a grid of equally spaced parallel lines distances a and b apart, with l<a,b.
Solution:
The position of the needle can be specified with points (x,y) and its orientation with coordinate ϕ. By symmetry, we can consider a single rectangle of the grid, so 0<x<y and x<y<b. In addition, since opposite orientations are equivalent, we can take –π/2<ϕ<π/2.
Now, the probability for any standard polygon can be given by,
[Notice how the denominator has an irrational number: π]
Integrating, using Laplace’s derivation for squares, the general solution is
If a=b so that x=l/a=l/b and 0<x<1, the probabilities of respective needles crossing junctions are:
Now, for a modified case where the layout comprises of equilateral triangles of side length a, the solution is,
The type of probabilities obtained in this Buffon-Laplace Needle Problem, and its variants, is an excellent example of how Classical Theory of Probability fails to accommodate irrational numbers which prove to be essential for the real life-applications of Probability Theory. From fluctuations in stock exchange to predictions of physical phenomenon, the axiomatic theory serves as the unifying theory of probability. Therefore, it would be apt to address the issue of “irrationality” of classical probability through a deeper understanding of Kolmogorov’s probability.
