Some Basic Concepts Probability

Random Experiment
A random experiment is a process leading to two or more possible outcomes, with uncertainty as to which outcome will occur.

Examples of random experiments include the following:

1.    A coin is tossed and the outcome is either a head or a tail.
2.    A customer enters a store and either purchases a shirt or does not.
3.    The daily change in an index of stock market prices is observed.
4.    A six sided die is rolled.

In each of the random experiments listed we can specify the possible outcomes, defined as basic outcomes. For example, a customer either purchases a shirt or not.

Sample space

The possible outcomes of a random experiment are called the basic outcomes, and the set of all basic outcomes is called a sample space. The symbol S will be used to denote the sample space. 

Example: What is the sample space for the roll of a single six-sided die?
Solution: The basic outcomes are the six possible face numbers, and the sample space is

                     S = {1, 2, 3, 4, 5, 6}

The sample space contains six basic outcomes. No two outcomes can occur together, and one of the six must occur.

Example: An investor follows the Dow-Jones Industrial index. What are the possible basic outcomes at the close of the trading day?
Solution: The sample space for this experiment is

             S = [{1. The index will be higher than at yesterday’s close},
                     {2. The index will not be higher than at yesterday’s close}]

One of these two outcomes must occur. They cannot occur simultaneously. Thus, these two outcomes constitute a sample space.

Event

An event, E, is any subset of basic outcomes from the sample space. An event occurs if the random experiment results in one of its constituent basic outcomes. The null event represents the absence of a basic outcome and is denoted by.

Example: Suppose a coin is tossed twice. Let H and T denote the head  and tail of the coin respectively. Then the sample space  of the experiment is
                 S = {HH, HT, TH, TT}
 Let A be the event of head of the first coin, then A will contain the sample points A = {HH, HT}.

Mutually Exclusive Events

If A and B be two events, then they are said to be mutually exclusive if . That is, two events are said to be mutually exclusive if they have no common points.
Example: Suppose a coin is tossed twice. Let H and T denote the head  and tail of the coin respectively. Then the sample space of the experiment is
                 S = {HH, HT, TH, TT}
Let A be the event of head of the first coin and B be the event of tail of the first coin, then A and B  will contain the sample points A = {HH, HT} and B = {TH, TT}. Since A and B have no point in common, A and B are said to be mutually exclusive.

Union

Let A and B be two events in the sample space, S. Their union, denoted A U B  , is the set of all basic outcomes in S that belong to at least one of these two events. Hence, the union A U B occurs if and only if either A or B or both occur.

More generally, given the K events E1, E2,……….,Ek, their union,  is the set of all basic outcomes belonging to at least one of these K events. 

Collectively Exhaustive

If the union of several events covers the entire sample space, S, we say that these events are mutually collectively exhaustive. Since every basic outcome is in S, it follows that every outcome of the random experiment will be in at least one of these events. Given the K events E1, E2,…………,Ek in the sample space, S, if =S , these K events are said to be collectively exhaustive. For example, if a die is thrown, the events “Result is at least 3” and “Result is at most 5” are together collectively exhaustive.