On Nov 1, 10:46 pm, Chris Reed wrote:
Tom's calculations indicate how hard it is for the human mind to grasp
probability, and thus why we cannot calculate risk properly.

Tom's coin analogy fails because he is looking for an unbroken sequence
of survival,

Well, yes. If I die on Tuesday, it would seem somewhat
optimistic to assume that I'll be alive on Wednesday!
Am I missing something?

which therefore takes into account the past in predicting
the future.

Er, no. Conditional probability and all that!

His calculations are cumulative. Even with coin tosses, we
can see that once we ignore the past and stop cumulating results, the
calculation changes.

Er, for the conditions I stated, no.

Thus, at the start of the week, the chance of survival for a week at
coin toss levels is 1 in 128. The chance of surviving for 8 days is
worse, at 1 in 256. However, if our subject survives day 1, his chance
of making day 8 increases to 1 in 128, and by the end of day 7 it has
risen to 50:50.

True, but missing the point.

The chance of getting to day 7 from day 1 is 1 in 128, so
the chance of getting to day 8 from day 1 is still 1 in 256.
No change.

No one is disputing if you've reached day 7 then the
chance of getting to day 8 is 1 in 2. Conditional probability, etc.

But on day N the chance of getting to day N+356 is
vanishingly small.

The older he gets, the longer his chances of living
forever! I think (but as a European writing after what UK
government-sponsored has recently described as a "hazardous" level of
wine consumption I cannot be sure) this may be related to Zeno's paradox
(in Tom Stoppard's words, "... thus proving that the arrow never reaches
it's target and Saint Sebastian died of fright").

Rats. You took the words right out my mouth! Leibnitz
and Newton also had a few things to say in this area

If we ignore the past, however, each day's chance is the same at 0.5.
Thus Ray (may he live forever) is able to state that next year his
chances will be pretty much the same, if he makes it that far.

Quite correct. But of course we are actually talking about the
chance of him getting there (which would seem to be unfortunately
small based on his statements).

Cumulation of probabilities is what the human brain does automatically.
Suppose the chance of being killed on a glider flight is 1 in 1,000. The
mind (without extensive training) deals with this in a number of ways:

1. I can fly safely 999 times, then have to give up or I will certainly
die on the 1,000th. If I'm already dead, I was "statistically" unlucky.

2. I've had 500 flights, so my risk level has risen to 50:50.

3. At my club we fly 1,000 flights a year between us, so one of us is
sure to die flying.

Unless I'm badly mistaken, none of these are true statements.

Correct (except under pathologically perverse circumstances

I try to think as follows:

a. In the UK where I fly, gliding fatalities are on average around 2.5
per annum out of 5,000 pilots, so my "statistical" risk is around 1 in
2,000 of dying through gliding each year.

b. I can do a number of things to reduce my personal risk to less than 1
in 2,000, so I'll try to do those things.

c. This is, to me, an acceptable level of risk for the pleasure I get
from gliding.

The good thing is that these probabilities are not cumulative. I've been
flying for 11 years, so if they were cumulative my "statistical" risk
might be down to under 1 in 20. It ain't.

I think the concept of "cumulative" is seriously misleading
in this context.

I think what you're really trying to say is that the probability
of dying on day X from cause Y is *not independent* of the
probability of dying on day X+1 from cause Y.

Under such conditions the "1 in P^N" calculation is clearly
and simply invalid.

In the absence of other information, I chose to presume
"independent" and you have chosen "not independent".