Probability

Introduction:

The word probability (or chance) is commonly used in day today conversation.

For example, we make statements” possibly it will rain tomorrow”, “there is a good chance of getting the job” etc. the words ‘possibly’, likely’, etc. convey the sense that there is uncertainty about the happening of the event.

Example: tossing of a coin-there is a chance of getting head and tail.

Probability is defined as the ratio of the number of favourable cases to the total number of equally likely cases.

i.e. P(A)=

Note: Favourable number of cases and total number of cases are both positive whole n umbers. Favourable number of cases is less than or equal to the total number of cases.

0 ≤ p(A) ≤ 1

Definitions of terms:

Random experiment: Experiment that has two or more outcomes which vary in an unpredictable manner from trial to trial when conducted under uniform conditions.

Example: tossing of a coin , since it has 2 specified outcomes. But we are uncertain whether head will appear or tail.

 Sample point: Every decomposable outcome of a random experiment.

Example: when a coin is tossed, getting head is a sample point.

Sample space:  Set of possible outcomes of the random experiment.

Ex: when a coin is tossed sample space(S) is {H, T}

Event: One or more possible outcomes of an experiment. An event is a subset of a sample space.

Ex: when two coins are tossed getting two heads is an event.

An event may be simple or compound.

 It is simple if it corresponds to a single possible outcome of an experiment.

 Ex: getting two heads while tossing two coins.

The joint occurrence of two or more simple events is called compound event.

Ex:  getting one head when two coins are tossed since it is the joint occurrence of HT and TH.

Equally likely events: Two or more events are said to be equally likely if any one of them cannot be expected to occur in preference of the other.

Ex: getting 1 and getting 2 when a die is thrown.

Mutually exclusive: If two events A and B are said to mutually exclusive, then A ∩ B = ∅

Ex: getting head and tail.

Complementary events: If A^c is a complement of then A and A^c  are mutually exclusive. P(A) + P(A^C) = 1

Addition theorem: P(A ∪ B) = P(A)+P(B)-P(A ∩ B)

Conditional probability: P(A/B)=

Multiplication theorem: P(A ∩ B) = P(A).P(B)