Probability or Chance:

Probability of chance is a common term used in day-to-day life. For example, we generally say, ‘it may rain today. This statement has a certain uncertainty.

Probability is quantitative measure of the chance of occurrence of a particular event.

Experiment

An experiment is an operation which can produce well-defined outcomes.

Random Experiment

If all the possible outcomes of an experiment are known but the exact output cannot be predicted in advance, that experiment is called a random experiment.

Examples;

Tossing of a fair coin

When we toss a coin, the outcome will be either Head (H) or Tail (T)

Throwing an unbiased die

Die is a small cube used in games. It has six faces and each of the six faces shows a different number of dots from 1 to 6. Plural of die is dice.

When a die is thrown or rolled, the outcome is the number that appears on its upper face and it is a random integer from one to six, each value being equally likely.

Taking a ball randomly from a bag containing balls of different colors.

Sample Space

Sample Space is the set of all possible outcomes of an experiment. It is denoted by S.

Examples

When a coin is tossed, S = {H, T} where H = Head and T = Tail

When a dice is thrown, S = {1, 2 , 3, 4, 5, 6}

When two coins are tossed, S = {HH, HT, TH, TT} where H = Head and T = Tail

Event

Any subset of a Sample Space is an event. Events are generally denoted by capital letters A, B, C, D etc.

Examples

When a coin is tossed, outcome of getting head or tail is an event

When a die is rolled, outcome of getting 1 or 2 or 3 or 4 or 5 or 6 is an event

Equally Likely Events

Events are said to be equally likely if there is no preference for a particular event over the other.

Examples

When a coin is tossed, Head (H) or Tail is equally likely to occur.

When a dice is thrown, all the six faces (1, 2, 3, 4, 5, 6) are equally likely to occur.

Mutually Exclusive Events

Two or more than two events are said to be mutually exclusive if the occurrence of one of the events excludes the occurrence of the other

This can be better illustrated with the following examples

When a coin is tossed, we get either Head or Tail. Head and Tail cannot come simultaneously. Hence occurrence of Head and Tail are mutually exclusive events.

When a die is rolled, we get 1 or 2 or 3 or 4 or 5 or 6. All these faces cannot come simultaneously. Hence occurrences of particular faces when rolling a die are mutually exclusive events.

Note : If A and B are mutually exclusive events, A ∩ B = ϕ where ϕ represents empty set.

Consider a die is thrown and A be the event of getting 2 or 4 or 6 and B be the event of getting 4 or 5 or 6. Then

A = {2, 4, 6} and B = {4, 5, 6}

Here A ∩ B ≠ϕ. Hence A and B are not mutually exclusive events.

 Independent Events

Events can be said to be independent if the occurrence or non-occurrence of one event does not influence the occurrence or non-occurrence of the other.

Example : When a coin is tossed twice, the event of getting Tail(T) in the first toss and the event of getting Tail(T) in the second toss are independent events. This is because the occurrence of getting Tail(T) in any toss does not influence the occurrence of getting Tail(T) in the other toss.

Simple Events

In the case of simple events, we take the probability of occurrence of single events.

Examples

Probability of getting a Head (H) when a coin is tossed

Probability of getting 1 when a die is thrown

Compound Events

In the case of compound events, we take the probability of joint occurrence of two or more events.

Examples

When two coins are tossed, probability of getting a Head (H) in the first toss and getting a Tail (T) in the second toss.

Exhaustive Events

Exhaustive Event is the total number of all possible outcomes of an experiment.

Examples

When a coin is tossed, we get either Head or Tail. Hence there are 2 exhaustive events.

When two coins are tossed, the possible outcomes are (H, H), (H, T), (T, H), (T, T). Hence there are 4 (=2²) exhaustive events.

When a dice is thrown, we get 1 or 2 or 3 or 4 or 5 or 6. Hence there are 6 exhaustive events.

 Algebra of Events

Let A and B are two events with sample space S. Then

A ∪ B is the event that either A or B or Both occur. (i.e., at least one of A or B occurs)

A ∩ B is the event that both A and B occur

A¯ is the event that A does not occur

A¯ ∩ B¯ is the event that none of A and B occurs

Example : Consider a die is thrown , A be the event of getting 2 or 4 or 6 and B be the event of getting 4 or 5 or 6. Then

A = {2, 4, 6} and B = {4, 5, 6}

A ∪ B = {2, 4, 5, 6}

A ∩ B = {4, 6}

A¯ = {1, 3, 5}

B¯ = {1, 2, 3}

A¯∩B¯ = {1,3}

Probability of an Event

Let E be an event and S be the sample space. Then probability of the event E can be defined as

P(E)= n(E)/n(S)

where P(E) = Probability of the event E, n(E) = number of ways in which the event can occur and n(S) = Total number of outcomes possible.