What is a Statistical Experiment?

All statistical experiments have three things in common:

• The experiment can have more than one possible outcome.
• Each possible outcome can be specified in advance.
• The outcome of the experiment depends on chance.

A coin toss has all the attributes of a statistical experiment. There is more than one possible outcome. We can specify each possible outcome (i.e., heads or tails) in advance. And there is an element of chance, since the outcome is uncertain.

The Sample Space

• A sample space is a set of elements that represents all possible outcomes of a statistical experiment.
• A sample point is an element of a sample space.
• An event is a subset of a sample space - one or more sample points.

Probability of an Event

With some statistical experiments, each sample point is equally likely to occur. In this situation, the probability of an event is very easy to compute. It is:

P(E) =	Number of sample points in event
	Number of sample points in sample space

Think about the toss of a single die. The sample space consists of six possible outcomes (1, 2, 3, 4, 5, and 6). And each outcome is equally likely to occur. Suppose we defined Event A to be the die landing on an odd number. There are three odd numbers (1, 3, and 5). So, the probability of Event A would be 3/6 or 0.5.

Types of events

• Two events are mutually exclusive if they have no sample points in common.
• Two events are independent when the occurrence of one does not affect the probability of the occurrence of the other.

Test Your Understanding

1. Suppose I roll a die. Is that a statistical experiment?
   
   Yes. Like a coin toss, rolling dice is a statistical experiment. There is more than one possible outcome. We can specify each possible outcome in advance. And there is an element of chance.
   
   
2. When you roll a single die, what is the sample space?
   
   The sample space is all of the possible outcomes - an integer between 1 and 6.
   
   
3. Which of the following are sample points when you roll a die - 3, 6, and 9?
   
   The numbers 3 and 6 are sample points, because they are in the sample space. The number 9 is not a sample point, since it is outside the sample space; with one die, the largest number that you can roll is 6.
   
   
4. Which of the following sets represent an event when you roll a die?
   
   A.   {1}
   B.   {2, 4,}
   C.   {2, 4, 6}
   D.   All of the above
   
   The correct answer is D. Remember that an event is a subset of a sample space. The sample space is any integer from 1 to 6. Each of the sets shown above is a subset of the sample space, so each represents an event.
   
   
5. Consider the events listed below. Which are mutually exclusive?
   
   A.   {1}
   B.   {2, 4,}
   C.   {2, 4, 6}
   
   Two events are mutually exclusive, if they have no sample points in common. Events A and B are mutually exclusive, and Events A and C are mutually exclusive; since they have no points in common. Events B and C have common sample points, so they are not mutually exclusive.
   
   
6. Suppose you roll a die two times. Is each roll of the die an independent event?
   
   Yes. Two events are independent when the occurrence of one has no effect on the probability of the occurrence of the other. Neither roll of the die affects the outcome of the other roll; so each roll of the die is independent.