Introduction to Probability Theory

Probability

A probability refers to the percentage chance that something will or will not happen. Probability of any event is a number between 0 and 1. If we are sure that the event will occur, we say that its probability is 100% or 1. But if we are sure that the event will not occur, we say that its probability is 0. Note that the probability measures the amount of uncertainty. The concept of probability is required for understanding of communication systems and many other real-life subjects.

Random Experiment

A random experiment is an experiment or a process for which the outcome (result) cannot be predicted with certainty.

It is a process by which we observe something not known (uncertain). For example, before rolling die we do not know the result. After the experiment is conducted, the result is known. Here the ‘result’ is called an OUTCOME of an experiment. The set of all possible results is called the sample space.

The word random means unpredictable, uncertain, in-deterministic. We call random because it can take several different possible values.

• Note that the outcome of a random experiment cannot be predicted with certainty, before the experiment is run.
• Note also that set of all possible results are known. In a coin toss experiment set of results = {H, T}.
• The experiment can be repeated any number of times
• The result of an experiment is not known before the experiment is run

Sample Space: Set of all possible outcomes (results).

Sample point: Each outcome of a sample space is known as sample point. It is also called element of a sample space.

Event: A subset of the sample space. So an event may be a single element or set of elements of a sample space.

Random variable: A random variable is a set of possible values from a random experiment. A random variable can assume several set of possible values. These possible values are numerical outcomes of a random experiment. Note that random variable is a quantity, whose value is subject to change.

Some examples of random experiments and sample spaces are shown below:

1. Experiment: toss a coin

Sample space S = {heads, tails} or S = {H, T} or S = {1, 0}

2. Experiment: roll a die

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

3. Experiment: observe the number of ice creams sold in a restaurant

Sample space S = {1, 2, 3, …….}

4. Experiment: drawing a card from a pack of 52 cards

Sample space S = {2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K, A}

When a random experiment is repeated several times, each one of them is called a trail. In the example of tossing a coin, each trial will result in either heads or trails. It is important to note that the sample space is defined based on the behaviour of experiment.

Suppose, a random experiment is defined as ‘score shown on the face of die when it is rolled’

Here sample space S = {1, 2, 3, 4, 5, 6}

If we throw a fair die having 6 faces, the number of possible outcomes is: 6 {1, 2, 3, 4, 5, 6}. Note that the possible outcomes of any random experiment are assumed to be known. But actual result of an experiment is not known until the experiment is conducted.