Probability is about the odds that a given event (or set of events) will occur. The weatherperson might say that the probability of rain this afternoon is 10%, meaning that, in the long run, 10% of the days on which such a prediction is made will have some rain, and the other 90% will have no rain.
Any discussion of probability requires considering the set of possible outcomes or states or situations. We assume that exactly one state from a set occurs at any time. If there is a set of a fixed size, then the calculations are much easier, however, sometimes, the set is of infinite size such as the set of all possible temperatures between 0 and 100 degrees.
If the set 
is finite then there is a probability 
that
each member 
can occur, such that:
since one state must be true at any time.
If the set 
is infinite, then the probability of any specific
state is zero; rather,
we assign discrete probabilities to ranges of values from 
.
Assuming that the values in 
are continuous, then the
usual notion of probability is the probability that the variable
x is between two values:
where 
is the probability density that describes the
distribution of the likelihood that a given event or value of x occurs.
As with discrete probabilities, some event must occur so:
One common continuous distribution is the uniform distribution, which has all possible values being equally likely between the upper and lower bound of the possible values. If the range of possible values is from a to b, then the probability density is:
Another common continuous distribution is the gaussian or normal distribution,
which can have any possible value, but whose values are clustered about
a mean 
, and the degree to which they are clustered is determined
by the standard deviation parameter 
:
One often talks about events or measurements being independent. This means that the variations in the value of the first has no relation to the variations in the value of the second event or measurement. This allows us to calculate the probability of a pair of events A and B as:
We can also talk about conditional probabilities, which occurs
when events or measurements have a relationship.
This is often expressed as 
meaning the probability of
A given that B is true.
If A and B are independent, then 
.
Suppose that we measure a value A and know that this measurement
could arise from cause B or C.
Suppose that we would like to know what the probability that
cause B is the correct cause.
This is expressed 
.
It may be hard to calculate this directly.
Fortunately, there is a famous rule of probability called Bayes law
that says:
We might know these other distributions and values, so we can than
derive the desired 
.