Probability is a measure of the likelihood of some event happening.
(Just about anything can be an event in this sense.) We measure
on a scale from 0 to 1 (or 0% to 100%), where smaller numbers indicate
less likelihood and higher numbers indicate greater likelihood. This
system in an example of how mathematics is used to formalize and make
precise our informal notions about some things being more or less likely
to occur than other things.
Two kinds of probability
We can distinguish two kinds of probability. Mathematical probability
is the measure of the relative frequency of an event
occurring. In addition, we will use the term
personal probability for a statement of someone's degree
of belief that an event will occur.
Determining Relative Frequency (Mathematical Probability)
If the process under study can be repeated or simulated many times, we can
determine the empirical probability by keeping track of
the outcomes in our (large number of) trials. The probability assigened is
P(A happens) = (# times A happened) / (# trials)
If the number of trials is
very large, then it is quite likely that this will give us a reliable estimate.
Sometimes we can make mathemitical assumptions about a situation and use
Four Basic Properties of Probability to determine the
theoretical probability of an event. The accuracy of a
theoretical probability depends on the validity of the mathematical assumptions
The four useful rules of probability are:
Empirical probabilities will also follow these rules (for a given set
of trials). Becuase people often have a poor sense of the likelihood
of an event, personal probabilities often do not follow these rules.
A collection of personal probabilities is called coherent
if it does not violate the rules for mathematical probability.
- It happens or else it doesn't.
The probabilty of an event happening added the probability of it
not happing is always 1.
P(A happens) + P(A doen't happen) = 1
- Exclusivity. If A and B can't both happen at the
same time (in which case we say that A and B are mutually exclusive), then
P(either A or B happens) = P(A happens) + P(B happens)
If B is no more or less likely to happen when A happens than when A doesn't
(in which case we say that A and B are independent),
P(A and B both happen) = P(A happens) * P(B happens)
If whenever A happens B must also happen, then B must be at least as
likely as A, so
P(A happens) <= P(B happens)
Equally likely outcomes
One especially important use of these probability rules is the
conclusions that can be drawn if we assume that a number of events
are equally likely. If there are only n such events that are
possible in a given situation, and all are equally likely and pairwise
mutually exclusive (no two can happen at once), then
each must have probability 1/n. More complicated situations
can be handled by dividing a situation into a number of equally likely
outcomes and counting how many of them are "of interest" (in the event).
The probabilty then is given by (number of interest)/(total number),
just as in the case for empirical probability.
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Wednesday, 14-Jan-2004 07:21:01 EST
Department of Mathematics and Statistics