Topics Probability

Question Is an event which has zero probability of occurring but which is nonetheless conceivably possible rightly termed "impossible"? For instance, is it "impossible" that I could be the EXACT same height as another person? I take it that the chance of this is zero in that there are infinitely many heights I could be (6 ft, 6.01 ft, 6.001 ft, 6.0001 ft, etc.) but only one which could match that of a given other person exactly; at the same time, I have no problem at all imagining a world in which I really am exactly as tall as this other.

Accepted:
June 28, 2007

Comments

As you make finer and finer measurements in the way you suggest, the probability declines each time by a factor of 10. As you go on and on, it shrinks below any value no matter how small. But, no matter how long you go on, it will never be zero, it will always be more than zero. (This is analogous to how, when you count, you'd eventually surpass any number anyone cares to specify but never reach infinity.) OK, so far no paradox.

But mathematics also recognizes numbers whose decimal extensions are infinitely long. And if you express each person's height as such a number, then your paradox does indeed arise. And I am not surprised, as there are other paradoxes involving infinite numbers as well, for instance, that there are as many even numbers as there are natural numbers, as demonstrable by a one-to-one mapping. Still, this is a nice addition (at least to my stock)! Let's see what others think.

Probability theorists often consider random trials with infinitely many possible outcomes each with probability zero--for example, the probability that a quantum particle will be at some particular point in space. In such cases, the probability that the result falls within an (infinite) set of such outcomes need not be zero----the probability that the particle is in some region of space, say.

I don't see that there is anything paradoxical here. It's true that cases like this violate the 'additivity' assumption that the probability of a disjunction of non-overlapping outcomes is the sum of the probabilities of the individual outcomes. But there's nothing manadatory about this assumption when we are dealing with infinite sets of outcomes, and probability theories covering this kind of case are perfectly consistent.

The question asked whether we should use the term 'impossible' for probability zero outcomes like the particle being at some particular point in space. I'd say not, given that the particle has to be somewhere. Probability zero does not mean impossible for outcomes like these.

I agree that there's nothing paradoxical here; surprising, perhaps, but not paradoxical.

The only kind of additivity that is usually assumed in probability theory is countable additivity, and there's no violation of that here. But you do have uncountably many non-overlapping outcomes, each with probability zero, such that the probability of at least one of those outcomes happening is one. So uncountable additivity doesn't work.

I would agree that an outcome with probability zero need not be impossible. Consider, for example, flipping a coin infinitely many times. Each infinite sequence of heads and tails has probability zero of occurring, but one of them has to occur, so it wouldn't make sense to say that they're all impossible. (Notice that there are uncountably many possible sequences of heads and tails.)

But of course this is not a realistic experiment--no one can actually flip a coin infinitely many times. The original example proposed also seems unrealistic to me--according to the uncertainty principle, the individual protons, neutron, and electrons at the top of my head don't have precise positions, so I'm not sure it makes physical sense to think of my height as a precise real number. I'm not sure if there is any realistic example of a physical event with outcomes that are possible, but have probability zero.