Thinking out loud

Thursday, January 26, 2006

Probability - All in the mind?

It’s been a while since I have been troubled by the notion of probabilities; trouble with the simple notion that a coin showing up heads has a probability distribution. I have had a hard time trying to articulate what I have felt and I finally think I have succeeded somewhat. It is my contention that in a quantum certain world, probabilities have nothing to do with an experiment at all; probabilities are only a summary of the state of our knowledge about the world and its laws. I will try to justify this position.

We start with the simplifying assumption that the world is quantum certain. We’ll add quantum uncertainty and worry about its effects later. In a quantum certain universe, the current state of the universe completely determines the future states of the world (or universe). The connection of all the variables may be extremely complicated and may be affected by “a butterfly flapping its wings in Australia”; but the future is certainly determined by the present state of the universe. Now let us try out an experiment in such a universe. I have a biased coin. Initially I do not know what the “probability” of the coin showing up heads is. Without any prior information about the coin, I would say that the probability of the coin showing up heads is half. I then proceed to toss the coin and find that it does indeed show up heads. I repeat the experiment a few times and find that the coin shows up heads 8 times out of 10. This leads me to adjust my probability assessment of the coin showing up heads. Now I show the coin to a friend and ask him to make an assessment of the probability distribution. Again, without any prior information, my friend is going to put the probability at half. I then proceed to flip the coin again. Now here is the point to consider: what is the actual probability of the coin showing up heads – half? Or 0.8? Clearly, my friend’s or my belief should not affect the outcome of the toss at all. Indeed, in a quantum certain world, the end result of this experiment is already decided – only we don’t know what it is. So in this case probability is indeed only in our mind and it has nothing to do with the act of flipping the coin at all. It may be argued that probabilities should not be interpreted as being assigned to a single event but as the statistics arising out of several repetitions of an experiment. However, if the world is quantum certain, why should it be the case that asymptotically the coin will continue to show the same statistical probability of showing up heads 8 out of 10 times? Why may not the predetermined chain of events lead to a completely different statistical behavior in the long run?

Now, let us add quantum uncertainty. This clearly allows probabilities to be “out there” instead of being only a figment of our imagination. However, there is still a problem. Why is it that uncertainties at the level of the tiny atomic/sub-atomic particles so precisely affect and control the probabilities of much more coarse-grained events like coin flips and card selections? How do uncertainties at the quantum level lead to such probability distributions that can be so easily analyzed, like say the probability of picking a king of spades from a pack of 52 cards? Do they infact affect these events at all? Or are these probabilities (of gross events, not quantum events) still only creations of our imagination; a statement of our ignorance of the state of the world?

3 Comments:

  • Interesting blog. Let me try and resolve this the way I have understood and learnt probability.
    We have a set of events - called the event space. And we have a set of outcomes, which is a subset of the eventspace. Clearly every event you imagine, need not be a real outcome. Outcomes are what are real and tangible. Probability is simply a function (that is between [0,1] and has integral over entire outcomespace = 1) defined on this set of outcomes. This is what we call the probability of an outcome (loosely often called the probability of the event).
    There are several ways of defining this probability function, which inturn yield different values for probabilities of each outcome. The definitions change based on the amount of information one has about the outcomes under consideration. Hence each observer is free to define his or her own probability function. In case of REAL situtations which by definition do not allow you to have complete information a convinient way of arriving at probabilities is by doing it emperically - observing asymptotic behaviour. It is only for this reason what the "probability function", which is indeed a figment of our imagination, ends up resembling the real behaviour of outcomes.

    By Anonymous Ankur, at 7:34 AM  

  • Exactly. So then we should completely throw out the notion of prior probabilities. And even so, consider an urn with one white ball and one black ball. Even before drawing a single ball, we know that asymptotically the probability of drawing a white ball is 0.5. Where is the empirical input here. If we agree that this experiment has been done several times before, we can change the setting ever so slightly to make sure it has not been done before, certainly not to our knowledge. Maybe we can replace balls with coloured t-shirts or coloured ball-pens or alarm clocks. The probabilities will still be half and half. My problem then is that why is the number of parameters affecting these probabilities so small and these probabilities remain unaffected by so many other things.

    By Blogger Parijat, at 7:56 AM  

  • i noticed that u have a problem with probabilities and quantum mechanics (which of course concerned with a lot of other things outside probability) many times...but i never really understood what troubles u.
    Neway, u mention that when asked what the probability for heads to come up on a coin is, ur first guess wud be 0.5. I dont think u shud guess at all. The 0.5 is arrived at assuming both events are totally similar rite? So unless u have ne information about the coins u shudnt speak about their probabilities to do nething at all. So thats the input we are using.
    And the point u make about quantum uncertainties, i fail to see.

    By Blogger Prashant Pawan, at 11:08 AM  

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