Expectation:
What a person believes will happen in certain conditions; or, in
probabilitytheoretical based reasoning: the
product of probability and
value. 1. In the first sense,
expectation is the basis of habit and of much
of what one believes. Normally, expectations are expectations of
probabilities, which is to say that many though not all expectations are not
confident certainties.
It should be noted that expectations are personal and normally based on
learning, including false learning: What one expects may be in part based on
illusion, and indeed much of what people
expect is not based on their own experience, but on what they believe based on
what they were told by others ('hearsay evidence').
2. In the second sense, expectation as the
product of
probability and value, one has a useful if
also sometimes misleading idea to guide one's actions in so far as these are
guided by probabilities and seek to maximize one's
pleasure or make the most of one's ends.
Note that in the last sense, and in general, it is useful to explicitly
make probabilities and values personal: If a is the name of a person, one
accordingly has
'p(a,q)' = 'the probability a attributes to the event that q'
'v(a,q)' = 'the value a attributes to the event that q'
'e(a,q)' = 'the expectation of a for q' = 'p(a,q)*v(a,q)'
Here it is also supposed  as a matter of useful convenience  that values
may be rendered as positive or negative numbers, the higher and more positive
for the more valued, and the lower and more negative for the events one does
not like to happen, in proportion to one's dislike for them.
This also makes it sensible to put v(a,q)=0 for those things q that person
a is indifferent to.
Three problems with the notion of expectation as defined in the second
sense is that it gives rise to some paradoxes (St. Petersburg Paradox); that it is
a statistical and 'on average' notion; and that it does not include reference to
what one can afford.
That the expectation of person a is defined as p(a,q)*v(a,q) makes its
application in guiding one's life and choices a statistical affair, where one
in fact tries to judge whether the doing of a specific act q is sensible
according to one's own appraisals of the probability and value of q by
referring this particular q to similar cases of q, and one compares this
possible act also with alternative expectations for r, s, t etc.
The criterion of maximizing one's expectation then counsels one to
choose that expectation from all one's expectations that one considers that is
highest, on the theory that this will give one, according to one's own
estimates of probability and value, the highest expected outcome.
This may be useful and sensible where one has many similar cases to go by,
and especially so if these indeed are very similar, but it may be less helpful
when one's choice concerns something rather unique and important, like the
choice of a wife.
Also, part of the problem here is that one judges in terms of
probabilities, which makes one's expectations fractions of values, whereas
what one gets is either the full value or nothing, depending on whether one
succeeds or fails.
A related problem with expectations is that it is wise when considering one's
expectations to also consider what happens if one fails i.e. p(a,~q)*v(a,~q),
and indeed if one can afford to fail. You may be willing to gamble a
dollar with a chance of 1 in 10 to gain 10 dollars, but it may be unwise if it
is your last dollar. In that case it may make more sense to spend it on the
certainty of having a one dollar meal, even if this is less satisfactory than
winning ten dollars, if one does.
