Probability

 

Introduction

ü   Systematic study began in the seventeenth century

ü   Game theory and gambling

ü   involve in uncertainty case

Some terms

ü  Experiment

Process which when perform gives different results (tossing a coin, throwing a dice)

ü  Sample space

Set of all possible outcomes of random experiment is called sample

Example S = {1, 2, 3, 4, 5, 6} throwing a dice

        S = {H, T} tossing a coin

        S = {HH, HT, TH, TT} tossing two coins

        S = {BB, BG, GB, GG} having two child

Throwing two dice simultaneously, the sample space S is

S = {(1, 1), (1, 2), (1, 3), …,(6, 6)}, 36 points

Tossing three coins, the sample space S is given by

S = {HHH, HHT, …, TTT}, 8 points

ü  Events

          Subset of sample space

          For S = {1, 2, 3, 4, 5, 6}, some events are {1, 3, 5}, {2, 4, 6}, {2, 3, 5} etc. How many events can be formed ? 64 events

ü  Simple event

containing only one element

S = {1, 2, 3, 4, 5, 6}, some events are {1, 3, 5}, {2, 4, 6}, {2, 3, 5}

ü  compound event

containing the combination of two or more simple events

 

S = {HH, HT, TH, TT} tossing two coins

{HH, TT}, {HT, TH}

ü  Exhaustive events

Totality contains all possible outcomes

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

ü  Equally likely events

Each has equal chance to occur

ü  Sure events

          contains all the possible outcomes (sample space)

ü  Impossible events

never occur, no chance to occur, f

ü  Mutually exclusive events

Two events are said to be mutually exclusive if they have no elements in common

A Ç B = f

ü  Complementary events

A Ç B = f and A È B = S

S = {1, 2, 3, 4, 5, 6}, some events are A = {1, 3, 5}, B = {2, 4, 6}

A Ç B = f

A È B = {1, 2, 3, 4, 5, 6} = S

Operation on events

ü Union

ü intersection

ü Difference

ü Complement [ S - E = complement of event S]

 

·    Classical or Mathematical definition of Probability: If there are n mutually exclusive and equally likely cases and m of them are favorable to certain event A, then the probability of the occurrence of event A is defined as the ratio m/n .

Finite and infinite sample space

If a sample space has finite points, then it is called finite sample space otherwise it is called infinite sample space.

If S = {a₁, a₂, a₃, …, a_n} with probabilities p₁, p₂, p₃, …, p_n, then

a) p_i ³ 0

b) p₁ + p₂ + p₃ + … + p_n = 1

Complementary Events

If A be any event of sample space S and A' be the complementary event of A, then

            P(A) + P(A') = 1

            or, P(A) =1 -  P(A') and P(A') = 1-  P(A)

Permutation

The different arrangements that can be made out of a given set of things by talking some or all of them are called their permutations.

Permutation of n things taken r at a time is denoted by P(n, r) or ⁿp_r.

The total number of a set of n objects taken r at a time is given by