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How meaningful are probabilities of single events?
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(Note: In this FAQ, 'event' is always understood in the ordinary sense
of the word, as 'something specific happening'.
In axiomatic probability theory based on Kolmogorov's axioms,
there is a slightly different, formal meaning of an event as an
element of the underlying sigma algebra.
An axiomatic foundation of probability theory equivalent to that of
Kolmogorov, but not based on sigma algebras, can be found in the book
'probability via expectation' by Paul Whittle, and a quantum extension
in quant-ph/0303047.)
Probabilities of single events are not at all meaningful
- at least not in any scientific sense -, although we are
used to scientific-sounding phrases such as
''There is a 60% probability for rain tomorrow''.
Instead, probabilities are properties of ensembles of events.
In the case just cited, the ensemble is the set of all tomorrow's,
(or rather an infinite idealization of it), and the probability is not
an exact probability, but an estimate computed on the basis of a sample
of former 'tomorrow's, together with statistical weather models.
(See the FAQ entry on ''Statistics of single systems''.)
Probability assignments to single events can be neither verified nor
falsified. Indeed, suppose we intend to throw a coin exactly once.
Person A claims 'the probability of the coin coming out head is 50%'.
Person B claims 'the probability of the coin coming out head is 20%'.
Person C claims 'the probability of the coin coming out head is 80%'.
Now we throw the coin and find 'head'. Who was right? It is undecidable.
Thus there cannot be objective content in the statement
'the probability of the coin coming out head is p', when applied to
a single case. Subjectively, of course, every person may feel
(and is entitled to feel) right about their probability assignment.
But for use in science, such a subjective view (where everyone is right,
no matter which statement was made) is completely useless.
What is the probability that a particular person, Mrs. X, will die of
cancer? This is a single event that either will happen, or will not
happen. If one considers this single event only, the probability is 1
or 0, depending on what will actually happen. (But this sort of
probability is not what we talk about in physics.)
On the other hand one may assign a probability based on some facts
about Mrs. X (smoker? age? gender? already ill?, etc).
Each collection of such facts determine an ensemble of people,
from which one can form a statistical estimate of the probability.
It clearly depends on which sort of ensemble one regarde Mrs. X
to belong to, what probability one will assign. Mrs. X belongs to many
ensembles, and the answer is different for each of these.
Thus probabilities are meaningful not as a property of the single event
but only as a property of the ensemble under consideration.
This can also be seen from the mathematical foundations. Classical
probabilities are determined by measures over some sigma algebra.
All statements in measure theory are _only_ about expectations and
probabilities of all possible (often infinitely many) realizations
simultaneously, and say nothing at all about any particular
realization.
For a random sequence consisting of 9 independent bits, with 0 and 1
equally likely, the sequence 111111111 has exactly the same status
and exactly the same probability as the sequences 110100101 or
000000000, although only the second sequence looks random.
(A random sequence is _not_ a sequence of numbers but a sequence of
random numbers = measurable functions. Only the _realizations_ of a
random sequence are sequences of ordinary numbers. Sequences of
ordinary numbers are _never_ random, but they can 'look random',
in a subjective sense.)