[Jim Noble]

The probability goes way up when it has been 760,000 years since the last one. Who knows that math?

I do, and you're wrong.

If you flip a coin 1000 times and it comes up heads every time, the probability of heads on the 1001st flip is 0.5.

[AaronInCarolina]

I do, and you're wrong.

If you flip a coin 1000 times and it comes up heads every time, the probability of heads on the 1001st flip is 0.5.

Jim, you are correct, theoretically. In an unchanging system for which a probability has been established (such as a coin that has not been weighted to favor one outcome versus the other) probability on any given event is always the same without regard to previous outcomes. However, in the scenario which you cited where a coin flip yielded heads every time for 1000 times, it is highly likely that the coin is not a "legal" coin (one which has a 1-in-2 chance of heads), and therefore the 1001st flip will highly likely yield another head. However, I know you were talking hypothetically about 1000 heads in a row with a legitimate coin (as unlikely as it is), and you are therefore quite correct about the 1001st flip chance of heads being 50% chance.

That being said, applying this analogy to a volcano, which presumably is an ever-changing system, in my opinion is incorrect. The article pointed out that the 1-in-740000 chance was a quite crude approximation based only upon the lapse of time since the last eruption. It is probably incorrect to apply such rigid standards (as one does to a legitimate coin) to a crude probability concerning a volcano. A volcano eruption is typically cyclic, and is therefore a constantly changing system whereby passage of time almost certainly increases the real probability of eruption, crude estimates not withstanding.

[Cboldt]

If you flip a coin 1000 times and it comes up heads every time, the probability of heads on the 1001st flip is 0.5.

A better analogy would be a part subjected to cyclical stress greater than its fatigue limit. Picking numbers of cycles at random, a part might be unlikely to fail in the first 10 million cycles, more likely to fail by 100 million cycles, and certain to fail before 500 million cycles. The probability of failure increases as the number of cycles increases.

Or, a boiler without a definite point of relief, subjected to a source of heat. Probability of failure is low while the boiler is cold, and increases as the temperature and pressure increase.

Probabalisitic design is a cool subject. Beats "safety factor" hands down.