Chance and Probability.


Take a small glass container. Half fill it with a white powder then, very carefully, add precisely the same amount of black powder. Examine the contents and note that, as expected, there are two layers, one white and one black. Now place a tight lid on the container and shake it vigorously. After the dust has settled within the container examine it again and note that it now contains what appears to be a grey powder. Shake it again and you will obtain the same result, it remains grey. The question is, could it ever be shaken back to its original state of one half black and one half white?

We may fairly safely say that this has never happened in human experience, the powder remains as a grey mixture no matter how many times it is shaken.

But supposing we take a very tiny container, so small that it will only hold a column of 4 grains of the dust. Now put in 2 white grains and then 2 black grains like this.
                                                                     
In this initial state, as in the larger jar, there are two distinct layers. If we now remove and scramble the four grains and put them back into the container we obtain one of the following 6 distribution possibilities:
                                                                                                                                               
The 4 grains in no way differ from those in the original glass container except in number, but here there clearly is a 1 in 6 chance that the grains will fall into their original state, and a 1 in 3 chance that they will recombine into 2 distinct groups as in the first and last outcome.

The probability can be ascertained mathematically for any number of grains; the number of combinations for n objects taken r at a time is n!/r!(n-r)!, indicated conventionally as nCr. In this simple case 4󫢪1 divided by 2x1x(4-2)! or simplified further 24/4 = 6. In the second case, where we are not bothered whether the black grains are at the top of the container or the bottom, we can halve that result giving us 1 in 3 chances.

Our value for r in the illustrated example was 2, i.e., half the total number of grains, since regardless of the individual grains half must always be at the top and half at the bottom. So r in our case is n/2 and the formula becomes n!/((n/2)(n-n/2)!).

For 200 grains (100 white and 100 black) the number of combinations is 200C100 = 9.054851465621034, which in full is 90, 548, 514, 656, 200, 000, 000, 000, 000, 000, 000, 000 of which only one possibility out of this vast number will get us back to the original order.

There are of course millions of dust particles in a few ounces of dust; as for molecules, suffice it to say that they are measured in moles and that the number of molecules in a mole is 6.022521023. It would require a hefty mainframe computer some considerable time to calculate the possible combinations of just one mole of a substance. And there are many moles in even small amounts of any substance; for example, 1 mole of H2O molecules weighs just 18.02 grams, a mere spoonful!

From this we can also be certain that there isn't even a remote possibility that time runs backward, as some science fiction writers have postulated, for if it did just a few shakes would get us from the mixed grey powder (the end product) to the initial two separated white and black layers (the starting position).

Statistically it is highly improbable that tepid water will separate spontaneously into one hotter and one colder portion, or that all the molecules of air will congregate in one corner of a room, but it is not impossible. A group of molecules might, in the course of their incessant colliding, sort themselves out, just once in a blue moon, into a hot fast group at one end of a container and a cool slow group at the other, or arrange themselves into a self-replicating organism.

If some sort of similar event triggered life on Earth then life could be very very rare indeed; it would not suffice to have any number of planets like the Earth in any number of galaxies. It would also require the fortuitous combination of the right molecules on each of those planets. Because an extremely unlikely event occurred once we should not assume that there is a certainty that it will regularly occur again; we may well be the only intelligent form of life in the universe.

Acknowledgements by way of a postscript

This is a bit like probabilities and chance in itself. I remembered reading about probability and the grey powder years ago, but I couldn't find the source and finally gave up. Some weeks later I happened to be searching for an article in Science & Belief from Darwin to Einstein (OU Press, 1980) and there it was: an extract from Physico-Chemical Evolution, published in 1919 by the Swiss physicist Eúgene Guye, an important figure in the early 20th century debate on life and thermodynamics. To quote Guye,

It will now be understood why the phenomenon only evolves in one directon and why it is irreversible. If the separation of a grey powder into its constituents does not occur when it is agitated it is not because the phenomenon is impossible but because it is only very slightly probable.
Guye concluded with a quotation from Herodotus,
If one is sufficiently lavish with time, everything is possible.
So perhaps I'm wrong in my speculation and, given time and the right conditions, life will spread throughout the universe! Meanwhile keep shaking that jar of grey powder, you never know!

Peter Ghiringhelli, B.A.(Hons), M.A.,
from an article I wrote for the Bradford Astronomical Society.

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