PROBABILITY- in 1993 A.N. Kolmogrov, a Russian mathematician tried succesfully to relate the theory of probability eith the set theory by axiomatic aproch.

Random variables - A real valued function defined on a sample -space is called a randon - variable .

SAMPLE SPACE- A sample space of a random experiment is the set of all possible outcomes of that experiment and is denoted by S.

for example- if a coin is tossed then there are two possibilities either we shall get a head or tail . we denoted here head (H) and tail(T).

EXHAUSTIVE EVENTS- All possible outcomes in a trial are called exhaustive events.

SAMPLE POINT:- Every element of the sample space is called a sample point. and it it is contains finite number of point then its is called finite sample point.for example in a coin H and T.

EVENTS:- Of all the possible outcomes in the sample space of an experiment some outcomes satisfy a specified description , it is called an event and its denoted by E.

CERTAINS AND IMPOSSIBLE EVENTS:- if S is a sample space then S and Φ are both subset of S and both are events.so S is called certains events and Φ is called impossible events.

EQUALLY LIKELY EVENTS:- Two wvents are considered equally likely if one of them cannot ve expected in preference to the other.

EXHAUSTIVE EVENT:- All possible outcomes in a trial are called exhaustive events.

for example :- if we trail a coin then the exhaustive events are H and T.

MUTUALLY EVENT OR INCOMPITIBLE EVENT:- Two or more than two evetns are called Mutually exclusive events if there are no element common to these events .

if E1 and E2 are two mutually exclusive events then E1 =E2 =Φ

EXAMPLES :- An experiment in which two coins are tossed together fine the sample space.

solve- S={(H,H) (H,T) (T,T) (T,H)}

IF coins tossed tree times then S ={ (HHH, TTT, THH, HHT, HTT, THT,}

SAMPLE OF COMPOUND EVENTS:- If E contains only one elements of the sample space S then E is called simple events i.e E= {ei}

compound events :- If E contains more than one elements of the sample space S then S is called compound event.

E={ei} where i = 1,2,3....n

FORMULLE FOR PROBABILITY OF AN EVENT:- let E be an event of S containing m element of S ie n(E) = m if P(E) is the probability of the event E happening then P(E) = n(E) /n(S)

COMPLEMENTARY EVENT:- if E be an event then not happining of the event E is called the complementary event of E and is denoted by E'.

and p(E) + p(E') = 1

Q:- find the probability of throwing on even number with a die.

solve :- let S be a sample space and the event of getting an event number be E then

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

so n(S) = 6 and n(E) = 3 [ for even number]

therefore , P(E) = n(E) /n(S)

= 3/6

= 1/2

Q:- if two coin tossed , find the chance that there should be heads on both.

solve:- if two coins tossed then sample space is

S={(H,H) (H,T) (T,T) (T,H)}

both head have only one condition that is ( H,H)

therefore P(E) = n(E) /n(S)

= 1/4

Random variables - A real valued function defined on a sample -space is called a randon - variable .

SAMPLE SPACE- A sample space of a random experiment is the set of all possible outcomes of that experiment and is denoted by S.

for example- if a coin is tossed then there are two possibilities either we shall get a head or tail . we denoted here head (H) and tail(T).

EXHAUSTIVE EVENTS- All possible outcomes in a trial are called exhaustive events.

SAMPLE POINT:- Every element of the sample space is called a sample point. and it it is contains finite number of point then its is called finite sample point.for example in a coin H and T.

EVENTS:- Of all the possible outcomes in the sample space of an experiment some outcomes satisfy a specified description , it is called an event and its denoted by E.

CERTAINS AND IMPOSSIBLE EVENTS:- if S is a sample space then S and Φ are both subset of S and both are events.so S is called certains events and Φ is called impossible events.

EQUALLY LIKELY EVENTS:- Two wvents are considered equally likely if one of them cannot ve expected in preference to the other.

EXHAUSTIVE EVENT:- All possible outcomes in a trial are called exhaustive events.

for example :- if we trail a coin then the exhaustive events are H and T.

MUTUALLY EVENT OR INCOMPITIBLE EVENT:- Two or more than two evetns are called Mutually exclusive events if there are no element common to these events .

if E1 and E2 are two mutually exclusive events then E1 =E2 =Φ

EXAMPLES :- An experiment in which two coins are tossed together fine the sample space.

solve- S={(H,H) (H,T) (T,T) (T,H)}

IF coins tossed tree times then S ={ (HHH, TTT, THH, HHT, HTT, THT,}

SAMPLE OF COMPOUND EVENTS:- If E contains only one elements of the sample space S then E is called simple events i.e E= {ei}

compound events :- If E contains more than one elements of the sample space S then S is called compound event.

E={ei} where i = 1,2,3....n

FORMULLE FOR PROBABILITY OF AN EVENT:- let E be an event of S containing m element of S ie n(E) = m if P(E) is the probability of the event E happening then P(E) = n(E) /n(S)

COMPLEMENTARY EVENT:- if E be an event then not happining of the event E is called the complementary event of E and is denoted by E'.

and p(E) + p(E') = 1

Q:- find the probability of throwing on even number with a die.

solve :- let S be a sample space and the event of getting an event number be E then

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

so n(S) = 6 and n(E) = 3 [ for even number]

therefore , P(E) = n(E) /n(S)

= 3/6

= 1/2

Q:- if two coin tossed , find the chance that there should be heads on both.

solve:- if two coins tossed then sample space is

S={(H,H) (H,T) (T,T) (T,H)}

both head have only one condition that is ( H,H)

therefore P(E) = n(E) /n(S)

= 1/4