Precarious probabilities...

The following is the excerpt from from the Preface of Roger Penrose's  "The shadows of the mind." It raised a question in my head that I had at first ignored because I had thought like the father (as you will see) and then a few months down the lane, a friend of mine pounced back at me with the daughter's question. And because my friend was not a three year old, i could brush her off ever so easily and i really had to think about it. 

In fact, it was a few days of searching and rummaging through a whole lot of books before I could spot this passage which was my first introduction to this question in probability (I am tempted to call it a paradox but my lack of aptitude on a subject does not merit it to be a paradox).

Here is the passage :

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Jessica always felt slightly nervous when she entered this part of the cave. "Daddy ? Suppose the great boulder fell down from where it is wedged between those other rocks. Wouldn't it block our way out, and we' never, ever, ever get home again?"

"It would, but it won't", replied her father a little distractedly, and somewhat unnecessarily brusquely, as he seemed more interested in how his various plant samples were accustoming themselves to the dank and dark conditions in this, the most remote corner of the cave.

"But, how do you know it won't, Daddy?" Jessica persisted.

"That boulder's probably been there for many thousands of years. It's not going to come down just we're here."

Jessica was not at all happy with this. "Surely, if it's going to fall down sometime, then the longer it's been there, the more likely it's going to fall down now?"

Jessica's father stopped prodding at his plants and looked at Jessica, with a faint smile on his face. "No, it's not like that at all.". His smile became more noticeable, but now more inward. "Actually, you could even say that the longer it's been there, the less likely that it's going to fall down when we're here." No further explanation was evidently forthcoming, and he turned his attention back to his plants.

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So who is right in your opinion ? Think about it. 

In my humble and ignorant opinion, this is a fascinating question and I have bugged a friend or two about this without a definitive answer. I can't say, I haven't made any headway at all but its definitely no where close to the resolution, I would have desired. 

Think about the question itself : What happens to the probability of an event if it has not happened in a long, long time - like the collapsing of the caves or the destruction of the sun or the big bang ? Or of a active volcano becoming dormant or exploding again ? Does it imply that the chance of them happening only decreases as time goes by or does it mean that the chance of its occurrence is diminishing progressively? 

Now if you leave aside these big phenomena and look at the question of a coin toss : if you haven't had a 'head' in a series of coin tosses (with a fair coin, on a flat surface and tossed the right way etc etc) - then the chance of you getting a 'head' in the next toss is only increasing... right ? I mean, it is an inevitable event... when you toss a coin, there is a 50-50 outcome and if for whatever reason the results are skewed, it only means that they will even out sooner than later. 

Now from what I see there is a fundamental difference between the two kinds of phenomena we are examining. In the former scenario, we are talking of big phenomena where there is no statistic or no record of the event for us to predict. But when we talk of smaller phenomena like coin tosses, we know the options and the inevitability/predictability of it (again - given all fair conditions). Thus, what my head is able to see is that before we predict the probability of an event, we need to know the inevitability of the event. Is it certain that the caves will collapse ? To me it seems that the answer would depend on the time scales again. In a matter of a million years, perhaps, yes ! But in a matter of months - almost certainly, no ! 

I wonder if you see my predicament here. 

For you to know the probability of something, you need to know if it will happen for certain. It implies that in a grander scheme of things, you, a mortal, have a handle on the future. Isn't that a contradiction - an almost impossibility ? 

Now, I hope you don't think that this is a esoteric quest with no meaning in the real world. I can almost hear you say - "After all who is asking about the probability of a cave collapsing or a mountain rising or the sun burning out?". But now think about the events in the past few years. Think of the lawsuit filed against the scientists in Italy when the earthquake happened despite their assurances to the contrary. (http://www.nature.com/news/2011/110914/full/477264a.html) What would you say to that ? I personally thought it was ridiculous but then, hey, I am a scientist (atleast nothing else yet) ! At the same time, the scientists are in no position to rule out the earth quake too and so that is how it must be stated. How about the economists who are trying to predict the stock markets' rise and fall, the property boom and bust, the future of the euro, the dollar, the rupee, the ren  or the yen ? They are basing their calculations on market trends - short term and long terms and with not much success. How about the meteorologists trying to predict rainfall, melting of glaciers, droughts and floods, wind patterns and the likes ? 

Does the likelihood of a flood in Bangladesh increase because there hasn't been one in the past two years or does it decrease for that very reason ? 

I have thought about this for a while and bugged a couple of friends as I said, and although I get their drift, I am still not totally convinced. And the funny thing is, I don't think even they are ! 

So... leave me a word, if you find something to add to this. I confess to not being a mathematical prodigy. In fact, other than dealing with numbers with some degree of speed and efficiency for routine calculations, I lack any skill with the sense of abstraction that mathematics demands. I wish things were different but my childhood fear of symbols, numbers, theorems and proofs has not yet dissipated but then again - we gain some and we lose some. Right ?