Probability

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:

The Cummulative Distrubution Function P(a), in the discrete case is the sum of the density function, from 0 to a. In the continous case, the sum is replaced by an integral.

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: