Risk Latte - Playing with Probabilities - the “probable” and the “impossible”

Playing with Probabilities - the “probable” and the “impossible”

Rahul Bhattacharya
April 29, 2008

We talk about probability measure and probability estimates all the time. Whether one is an equity analyst, a trader, an economist, a loan officer, a regulator, an actuary or whatever, he or she is continuously playing with "probabilities" and being battered by it. The stock has a 25% chance of going up, the call option has a 30% chance of finishing in the money, the counter-party default probability is 1%, there is a 25% chance of Fed lowering the rate by 50 basis points and so on and so forth. Every one of us, day in and day out, is confronted with and battered by these probability measures.

However, how often do we encounter a probability of "zero"? How often do we come across a situation when we say "this is impossible" or how often do we encounter hypotheses such as, "every single policyholder in a city dies" or "HSBC becomes bankrupt overnight". How often do we encounter a situation when ten aircrafts flying from the same airport crashes in mid-air on the same day or every single hedge fund in a portfolio of fund of funds drops 30% on the same day?

In other words, how often do you encounter situations with zero or "near zero" probabilities? Something that is "impossible". But what is a "zero" probability event? How do we define such an event?

In fact, it could be an immensely challenging task to define a zero probability event or a walk in the park, depending on how we look at it. The challenge really is defining a zero probability measure using the frequentist measure used in Statistics. You've never seen a live (real life) dinosaur in flesh and blood, so the chance of your encountering one on one of your hiking expedition is nil, zero. But does it mean that just because you've not encountered a live dinosaur (in flesh and blood) ever in your life, real life dinosaurs don't exist?

You look at history of events and then estimate a statistically significant measure from observing their occurrence in the past. But what if an event has no history at all? Does it mean that the probability of occurrence of that event is zero? On this we will not go any further, let's leave it for another column.

However, in an interesting article on the net* we recently came across a formulation that may well define "zero" probability event and yet keep us well within the frequentist approach of probability measurement. Here again, we start with a very small measure (truly small) and say that any measure smaller than that will signify a zero probablility. In other words, any chance (probability) measure smaller than the chosen benchmark (another very small measure) will have a statistical significance of zero.

The author of the same article draws inference from the world of sub-atomic particles in physics and the age of our universe. His arguments are:

  • There are 1084 sub-atomic particles in the known physical cosmos;
  • There are a maximum of 1020 interactions (oscillations/cycles) per second between any two of those sub-atomic particles;
  • There are 1017 seconds in the supposed age of the cosmos (15 billion years);

Take all three numbers above and multiply them to get a number 10121  (1084 x 1020 x 1017 = 10121 ). This number is an extremely large number and represents the total number of sub-atomic interactions possible since the beginning of our universe (the Big Bang). Now multiply this number by 10,000 and you get a slightly bigger number which will represent the total number of interactions of sub-atomic particles in ten thousand such universes. And this, according to the author, is "Cosmic Limit Law of Chance" or the benchmark for deciding zero probability event.

Therefore, any event whose chance (read probability) of occurrence is less than one in 10125 will be virtually an "impossible" event, i.e. have statistically the same significance as a zero probability event. And this will be true not just in our universe but ten thousand such universes.

Thus a probability of one in 10125 (i.e. 10-125 ) is deemed to be a "zero" probability event. How small is this number? Extremely small, so small that it is practically equal to zero.

Now, according to the above logic or formulism what would be an absolutely certain event? That is an event whose probability is one?

* http://www.geocities.com/Athens/Aegean/8830/mathproofcreat.html

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