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
Q:- Find the probability of throwing on even number with die.
solve:-let S be the sample space and the event of getting an even number be E then
S = {1,2,3,4,5,6} and E={2,4,6}
n(S) = 6 and n(E) = 3
therefore the probability of event of happening p(E)=n(E)/n(S)
= 3/6
=1/2 =ans
COMPOSITION OF EVENTS:-
THE EVENT REPRESENTED BY AUB OR A+B:-- IF the event E happens when A happens or B happens then E is denoted by and E is represented by AUB i.e. E=AUB
THE EVENT REPRESENTED BY A ∩ B OR AB:--If The event E happens when the events A and B both happens then the events E is represented by A ∩ B OR AB i.e. E=A ∩ B
COMPLEMENT OF EVENT A OR THE EVENT A':-- If the event E happpens when the event A does not happen then E is denoted by A'
Theoram :- If E1 and E2 are any two events then P(E1 U E2) = P(E1)+P(E2)-P(E1∩ E2).
PROOF:-
lets S be the sample space and n be the number of elements in the events in S.
let l be the number of elements in E1 and m the number of elements in the events E2
i.e. n(S)= n n(E2)= l n(E2)=m
if the events E1 and E2 are not mutually exclusive then the E1∩ E2 is not equal to phy.
let n(E1∩ E2) = r
clearly, n(E1U E2) = l+m-r
now the probability of E1 and E2 happening denoted by P(E1∩ E2) is given by
p(E1U E2) =
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
Q:- Find the probability of throwing on even number with die.
solve:-let S be the sample space and the event of getting an even number be E then
S = {1,2,3,4,5,6} and E={2,4,6}
n(S) = 6 and n(E) = 3
therefore the probability of event of happening p(E)=n(E)/n(S)
= 3/6
=1/2 =ans
COMPOSITION OF EVENTS:-
THE EVENT REPRESENTED BY AUB OR A+B:-- IF the event E happens when A happens or B happens then E is denoted by and E is represented by AUB i.e. E=AUB
PROOF:-
lets S be the sample space and n be the number of elements in the events in S.
let l be the number of elements in E1 and m the number of elements in the events E2
i.e. n(S)= n n(E2)= l n(E2)=m
if the events E1 and E2 are not mutually exclusive then the E1∩ E2 is not equal to phy.
let n(E1∩ E2) = r
clearly, n(E1U E2) = l+m-r
now the probability of E1 and E2 happening denoted by P(E1∩ E2) is given by
p(E1U E2) =
hence proved
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