Trials and Events

1. Sample Space and Events
1. Sample Space : The set of all possible outcomes of a trial.
2. Event : A subset of the sample space.
   
   
3. Elementary Event : A set in the sample space consisting of only one element.
4. Certain Event : An event that always occurs, i.e., the entire sample space.
5. Impossible Event : An event that never occurs, i.e., the empty set


2. Union, Intersection, Complement, and Mutually Exclusive Events
   For two events and in a sample space :
1. Union of and : The event where or occurs, .
2. Intersection of and : The event where both and occur simultaneously, .
3. Complement of : The event where does not occur, .
4. Mutually Exclusive Events: Two events and are mutually exclusive if they cannot both occur at the same time, i.e., .

Union	Intersection	Complement	Mutually Exclusive




3. Mathematical Probability and Statistical Probability
1. Mathematical Probability
   If all elementary events in a sample space are equally likely to occur, the probability of event occurring is given by:
   
   
   
   This is called mathematical probability.
2. Statistical Probability
   If a trial is repeated times and event occurs times, the relative frequency approaches a constant value as becomes very large. This value is called the statistical probability of event .


4. Basic Properties of Probability
   For any event in a sample space :
2. The probability of the certain event is (i.e., it always occurs).
3. The probability of the impossible event is (i.e., it never occurs).