** Next:** Combining Probabilities
** Up:** Basic Ideas
** Previous:** Basic Ideas

The simplest form of probability functions are discrete probabilities.
In this case, we have some variable which can take different
values or outcomes, .

**Examples:**
- if represents a statement such
as ``it will rain tomorrow'', it can take two values TRUE and FALSE;
if represents the result of a coin flip,
there would be the two possible outcomes, meaning heads and
meaning tails; if is the number drawn in the next lottery
draw, there would be 13,983,816 possibilities.

We denote the probability that takes a
particular value as . The probability must be between
and . If , takes the value with
certainty; if , definitely does not take the value ;
the closer is to , the greater the likelihood that
takes this value.
Assuming that can take only one of the values at a time (the
values are mutually exclusive), the probability that takes
or is . Since must take some value, the sum
of the the probabilities of all possibly values is . Thus, the
probability is * normalised*,

which means that always takes one of its allowed values.
We are often interested, however, in cases in which a random variable
takes a continuous range of values. In this case, the above must be
modified. We write a probability * distribution* or * density* as a function of
the continuous variable .

**Example: **
- A farmer might be
interested in the probability that a given amount of rainfall will
occur in a particular growing season. The rainfall, , could be
modelled by a continuous positive real number .

We cannot interpret as the
probability that takes the value , because that probability is
almost always going to be . You cannot have an infinite number of
possibilities with nonzero probability, because it could not be
normalised. For continuous distributions, is defined so that
the probability that
is between and is
, where
is arbitrarily small.
The normalisation condition converts from a sum to an integral, and
is,

** Next:** Combining Probabilities
** Up:** Basic Ideas
** Previous:** Basic Ideas
*Jon Shapiro*

*1999-09-23*