Probability:
Sample Space:
- The set of all possible outcomes of an experiment is called the sample space.
- It is denoted by ‘S’.
- Each outcome is called element of sample space or simply a sample point.
Example:-
1.Flipping/tossing a coin.
S={H,T},|S|=2
2.Tossing two coins at a time.
S={HH,TT,HT,TH},|S|=4
3.A coin and a dice both are tossed.
S={H1,H2,H3,H4,H5,H6,T1,T2,T3,T4,T5,T6},|S|=12
4.Rolling of two dice.
S={(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(2,1),(2,2),(2,3),(2,4),(2,5),(2,6),(3,1),(3,2),(3,3),(3,4),(3,5),(3,6),(4,1),(4,2),(4,3),(4,4),(4,5),(4,6),(5,1),(5,2),(5,3),(5,4),(5,5),(5,6),(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)} , |S|=36.
Event:
The subset of a sample space is called event.
Example:-
1.Getting a head in tossing of a coin.
S={H,T}
E={H}
2.Getting 1 or 6 in rolling of a dice.
S={1,2,3,4,5,6}
A={1,6}
3.Getting at least one tail in the tossing of two coins.
S={HH,TT,HT,TH}
B={HT}
Types of Events:
Universal/Sure/Certain Event:
An event which cover each and every sample point of a random experiment.
So here E=S
Example:-getting sum of at least 2 in the tossing of two dices.
So here E=S
Null/Impossible Event:
An event which doesn’t cover any sample point of a random experiment
E={ } or Φ
Example:-getting sum 13 in rolling of two dices.
So here E={ } or Φ
Elementary/Simple event:
An event containing only single sample point of a random experiment.
Example:-getting both heads in tossing of two coins
So here E={HH}
Compound/Composite Event:
An event containing two or more sample points of a random experiment.
Example:-getting one head in tossing of two coins
So here E={HT,TH}
Equally likely Events:
Events those have same chances of occurrence in an unbiased random experiment.
Example:-each face(head or tail) on a coin is equally likely to occur when the unbiased coin is tossed.
Each number on a dice is equally likely to occur when the unbiased dice is rolled.
Complementary Event:
An event which contain the sample point those are not favoring the event. It is denoted by A’/Ā/not A
So A′=S-A
Example:-not getting 2 heads in the tossing of 2coins
S={HH,HT,TH,TT}
A=event of getting 2 heads={HH}
A′=event of getting not 2 heads={HT,TH,TT}=S-A.
Mutually Exclusive/Disjoint Events:
Two events are said to be mutually exclusive or disjoint if they don’t have any common sample point between them. So,
A∩B={ } or Φ
Example:-In the tossing of dice
S={1,2,3,4,5,6}
A=event of getting even number then A={2,4,6}
B=event of getting odd number then B={1,3,5}
Here A∩B={ } or Φ
A and B are mutually exclusive.
Compatible Events:
Two events are said to be compatible, if they have some common sample points between them.
So, A∩B is not equal to { } or Φ
So, they are not mutually exclusive events.
Example:-In the rolling of 2 dices
|S|=36
A= Event of getting sum 8
So A={(6,2),(5,3),(4,4),(3,5),(2,6)}
B=event of getting a doublet
So B={(1,1),(2,2),(3,3),(4,4),(5,5),(6,6)}
Hence A∩B={(4,4)}
Exhaustive/Collectively exhaustive Events:
Any number of events will cover each and every sample point at least once. So E1∪E2∪E3….∪En=S
Example:-in the tossing of 3 coins S={HHH,HHT,HTH,THH,
HTT,THT,TTH,TTT}
A=Getting at least 2 heads={HHH,HHT,HTH,THH}
B=getting at least 2 tails={HTT,THT,TTH,TTT}
C=getting same on all coins={HHH,TTT}
A∪B∪C=S
Independent Events:
If happening or failure of one event will not affect the happening or failure of other event, then both events are said to be independent.
Example:-In the tossing of a coin and a dice together.
A=event of getting head on coin.
B=event of getting an odd on dice.
Example:- picking of sweet from a sweet box containing same sweet.
PRESENTED BY Lopamudra Parida( Lect. in Mathematics)
