Probability is the
mathematical study of uncertain events. Some fundamental definitions are
needed:
Experiment – An action whose results cannot be predicted with
certainty. Rolling a six-sided die is an example of an experiment.
Outcomes – The possible results of an experiment. An outcome
of rolling a die would be rolling a five.
Sample Space – The collection of all possible out comes of an
experiment. The sample space of rolling a six-sided die would contain 6
outcomes and could be listed out as the set {1, 2, 3, 4, 5, 6}.
Event – A sub collection of the out comes in the sample
space. An event would be "rolling an odd number". This event
contains the outcomes {1, 3, 5}.
Note that the event of rolling an odd number is an event one may be
interested in predicting. Further, this event is simply a set that
contains the outcomes from the sample space that satisfy "rolling an
odd number". In the experiment above, it is possible to count the
number of outcomes in the sample space and the number of outcomes in the
event. The ability to count the outcomes in sample spaces and events is an
important skill in probability.
When all the outcomes of an experiment are equally likely, the probability
of an event E is defined by:
Here P(E) stands for the probability of event E. Consider the
experiment of rolling a six-sided die. Let E be the event of rolling an
odd number. Then
The probability of event E can be thought of as the percentage of
possible outcomes that are contained in E. In this case 50% of the
outcomes lie in E.
Some basic facts about probabilities are:
P(E) = 0 means it is impossible for E to occur
P(E) = 1 means it is certain E will occur
The probability of event E will always be between 0 and 1.
The closer P(E) is to 0 the more unlikely the event and the
closer P(E) is to 1 the more likely the event.
