Probability, Definitions and Examples, Experiment, Sample Space, Event, Mutually Exclusive Events, Equally Likely Events, Exhaustive Events, Sure Event, Null Event, Complementary Event and Independent Events

Probability is a branch of statistics that measures the likelihood or chance of an event occurring. It helps in predicting the possibility of future outcomes based on available information. Probability is expressed as a number between 0 and 1, where 0 indicates an impossible event and 1 indicates a certain event. It is widely used in business, economics, finance, insurance, science, and everyday decision-making.

In simple terms, probability answers the question: “How likely is it that a particular event will happen?”

Definition

Probability may be defined as the numerical measure of the chance that a specific event will occur under given conditions.

1. Experiment

An experiment is a process or activity that leads to one or more possible outcomes.

• Example:

Tossing a coin, rolling a die, or drawing a card from a deck.

2. Sample Space

The sample space is the set of all possible outcomes of an experiment.

• Example:
  • For tossing a coin: S={Heads (H),Tails (T)}
  • For rolling a die: S={1,2,3,4,5,6}

3. Event

An event is a subset of the sample space. It represents one or more outcomes of interest.

• Example:
  • Rolling an even number on a die: E = {2,4,6}
  • Getting a head in a coin toss: E = {H}

4. Mutually Exclusive Events

Two or more events are mutually exclusive if they cannot occur simultaneously.

• Example:

Rolling a die and getting a 2 or a 3. Both outcomes cannot happen at the same time.

5. Equally Likely Events

Events are equally likely if each has the same probability of occurring.

• Example:

In a fair coin toss, getting heads (P = 0.5) and getting tails (P = 0.5) are equally likely.

6. Exhaustive Events

A set of events is exhaustive if it includes all possible outcomes of the sample space.

• Example:

In rolling a die: {1,2,3,4,5,6} is an exhaustive set of events.

7. Sure Event

A sure event is an event that is certain to occur. The probability of a sure event is 1.

• Example:

Getting a number less than or equal to 6 when rolling a standard die: P(E)=1.

8. Null Event

A null event (or impossible event) is an event that cannot occur. Its probability is 0.

• Example:

Rolling a 7 on a standard die: P(E)=0.

9. Complementary Event

The complementary event of A, denoted as A^c, includes all outcomes in the sample space that are not in A.

• Example:

If $A$ is rolling an even number ({2,4,6}, then A^c is rolling an odd number ({1,3,5}.

10. Independent Events

Two events are independent if the occurrence of one event does not affect the occurrence of the other.

• Example:

Tossing two coins: The outcome of the first toss does not affect the outcome of the second toss.