[Siegmund]

Our minds tend not to be well-equipped for the astronomical and the infinitesimal. Unusual things happen when both of these factors come into play.

Returning to the original question: yes, given a sufficiently large number of trials, anything that can happen with positive probability on one trial will eventually happen. On the other hand, given any fixed number of trials, you can find events so fantastically rare that it is almost impossible for them to occur within that time frame.

To take your golf example - at least on a short enough course that reaching the green in one shot is physically possible on every hole, there is some positive but small probability of a series of aces happening. (Say 1 in 1000 per hole, for the sake of argument - published estimates for pros on short holes range from 1500:1 to 4000:1. Less for amateurs or on long holes of course.)

Two consecutive holes in one is literally a one in a million (or one in a few million) chance, something you expect to make the news when it happens, but something you can expect to happen somewhere in the world reasonably often. A quick google search reveals it has indeed happened several times. There is argument about whether three in a row has ever happened. That's getting down to once in a lifetime or once in several lifetimes chances, but still not 'impossible'.

In real-life terms, bucking 1 in 10^54 or more odds to hit 18 in a row is getting up to the range where it takes an entire universe full of golfers playing for eons before it will ever happen. As long as you are talking about golf, you can call that "impossible" safely.

On the other hand, imagine you are interested in collisions between air molecules - interested in two particular types of collisions at very specific angles and speeds, so that collisions meeting your requirements happen exactly as often as winning Powerball 3 times in a row (1 in 3x10^24) or as hitting 18 holes in one (call it 1 in 10^54 for this example - but maybe really as bad as 1 in 10^70.)

At the pressures and temperatures at the surface of the earth, each molecule in the air bumps in to another one about 2 billion times a second. In just the bottom-most kilometer of the Earth's atmosphere, there are about 3.5x10^43 molecules bouncing around. The "as rare as eighteen consecutive holes in one" event, then, happens somewhere near the surface of the earth an average of once every couple minutes. The "3 in a row Powerball Winner" rare event happens about 10,000 times per second in every cubic centimeter of air, not so rare at all when compared with some of the things they build detectors for in particle physics.

I guess my point is that "rare" is a relative term. There are things you will see only a few times, things you are unlikely to ever see, things that noone is likely to ever see, things that are unlikely to have happened anywhere in the universe since the beginning of time.