Probability deals with the analysis of random phenomena. It is a way of assigning every event a value between zero and one, with the requirement that the event made up of all possible results is assigned a value of one.
Experiment
An operation which results in some well-defined outcomes is called an experiment.
Random Experiment
An experiment whose outcome cannot be predicted with certainty is called a random experiment. In other words, if an experiment is performed many times under similar conditions and the outcome of each time is not the same, then this experiment is called a random experiment.
Example:
A). Tossing of a fair coin
B). Throwing of an unbiased die
C). Drawing of a card from a well shuffled pack of 52 playing cards
B). Throwing of an unbiased die
C). Drawing of a card from a well shuffled pack of 52 playing cards
Sample Space
The set of all possible outcomes of a random experiment is called the sample space for that experiment. It is usually denoted by S.
Example:
A). When a die is thrown, any one of the numbers 1, 2, 3, 4, 5, 6 can come up.
Therefore, sample space:
S = {1, 2, 3, 4, 5, 6}
B). When a coin is tossed either a head or tail will come up, then the sample space w.r.t. the tossing of the coin is:
S = {H, T}
Therefore, sample space:
S = {1, 2, 3, 4, 5, 6}
B). When a coin is tossed either a head or tail will come up, then the sample space w.r.t. the tossing of the coin is:
S = {H, T}
Sample Point or Event Point
Each element of the sample spaces is called a sample point or an event point.
Example:
When a die is thrown, the sample space is S = {1, 2, 3, 4, 5, 6} where 1, 2, 3, 4, 5 and 6 are the sample points.Discrete Sample Space
A sample space S is called a discrete sample if S is a finite set.
Event
A subset of the sample space is called an event.
Problem of Events
Sample space S plays the same role as universal set for all problems related to the particular experiment.
(i). ϕ is also the subset of S and is an impossible Event.
(ii). S is also a subset of S which is called a sure event or a certain event.
(i). ϕ is also the subset of S and is an impossible Event.
(ii). S is also a subset of S which is called a sure event or a certain event.
Types of Events
A. Simple Event or Elementary Event
An event is called a Simple Event if it is a singleton subset of the sample space S.
S = {H, T}
Then A = {H} occurrence of head and
B = {T} occurrence of tail are called Simple events.
B). When two coins are tossed, then the sample space is
S = {(H,H); (H,T); (T,H); (T,T)}
Then A = {(H,T)} is the occurrence of head on 1st and tail on 2nd is called a Simple event.
Example:
A). When a coin is tossed, then the sample space isS = {H, T}
Then A = {H} occurrence of head and
B = {T} occurrence of tail are called Simple events.
B). When two coins are tossed, then the sample space is
S = {(H,H); (H,T); (T,H); (T,T)}
Then A = {(H,T)} is the occurrence of head on 1st and tail on 2nd is called a Simple event.
B. Mixed Event or Compound Event or Composite Event
A subset of the sample space S which contains more than one element is called a mixed event or when two or more events occur together, their joint occurrence is called a Compound Event.
S = {1, 2, 3, 4, 5, 6}
Then let A = {2, 4 6} is the event of occurrence of even and B = {1, 2, 4} is the event of occurrence of exponent of 2 are Mixed events.
Compound events are of two type:
(i). Independent Events, and
(ii). Dependent Events
Example:
When a dice is thrown, then the sample space isS = {1, 2, 3, 4, 5, 6}
Then let A = {2, 4 6} is the event of occurrence of even and B = {1, 2, 4} is the event of occurrence of exponent of 2 are Mixed events.
Compound events are of two type:
(i). Independent Events, and
(ii). Dependent Events
C. Equally Likely Events
Outcomes are said to be equally likely when we have no reason to believe that one is more likely to occur than the other.
Example:
When an unbiased die is thrown all the six faces 1, 2, 3, 4, 5, 6 are equally likely to come up.D. Exhaustive Events
A set of events is said to be exhaustive if one of them must necessarily happen every time the experiments is performed.
Example:
When a die is thrown events 1, 2, 3, 4, 5, 6 form an exhaustive set of events.Important:
We can say that the total number of elementary events of a random experiment is called the exhaustive number of cases.E. Mutually Exclusive Events
Two or more events are said to be mutually exclusive if one of them occurs, others cannot occur. Thus if two or more events are said to be mutually exclusive, if not two of them can occur together.
Hence A1,A2,A3,...,An are mutually exclusive if and only if Ai∩Aj=ϕ, for i≠j
B). When a die is thrown, the sample space is S = {1, 2, 3, 4, 5, 6}
Let A is an event of occurrence of number greater than 4 i.e., {5, 6}
B is an event of occurrence of an odd number {1, 3, 5}
C is an event of occurrence of an even number {2, 4, 6}
Here, events B and C are Mutually Exclusive but the event A and B or A and C are not Mutually Exclusive.
Hence A1,A2,A3,...,An are mutually exclusive if and only if Ai∩Aj=ϕ, for i≠j
Example:
A). When a coin is tossed the event of occurrence of a head and the event of occurrence of a tail are mutually exclusive events because we cannot have both head and tail at the same time.B). When a die is thrown, the sample space is S = {1, 2, 3, 4, 5, 6}
Let A is an event of occurrence of number greater than 4 i.e., {5, 6}
B is an event of occurrence of an odd number {1, 3, 5}
C is an event of occurrence of an even number {2, 4, 6}
Here, events B and C are Mutually Exclusive but the event A and B or A and C are not Mutually Exclusive.
F. Independent Events or Mutually Independent Events
Two or more event are said to be independent if occurrence or non-occurrence of any of them does not affect the probability of occurrence of or non-occurrence of their events.
Thus, two or more events are said to be independent if occurrence or non-occurrence of any of them does not influence the occurrence or non-occurrence of the other events.
Let A is the event of occurrence of a red ball in first draw.
B is the event of occurrence of a black ball in second draw.
Then probability of occurrence of B has not been affected if A occurs before B. As the ball has been replaced in the bag and once again we have to select one ball out of 5(3R + 2B) given balls for event B.
Thus, two or more events are said to be independent if occurrence or non-occurrence of any of them does not influence the occurrence or non-occurrence of the other events.
Example:
Let bag contains 3 Red and 2 Black balls. Two balls are drawn one by one with replacement.Let A is the event of occurrence of a red ball in first draw.
B is the event of occurrence of a black ball in second draw.
Then probability of occurrence of B has not been affected if A occurs before B. As the ball has been replaced in the bag and once again we have to select one ball out of 5(3R + 2B) given balls for event B.
Occurrence of an Event
For a random experiment, let E be an event
Let E = {a, b, c}. If the outcome of the experiment is either a or b or c then we say the event has occurred.
Sample Space: The outcomes of any type
Event: The outcomes of particular type
Let E = {a, b, c}. If the outcome of the experiment is either a or b or c then we say the event has occurred.
Sample Space: The outcomes of any type
Event: The outcomes of particular type
Probability of Occurrence of an Event
Let S be the same space, then the probability of occurrence of an event E is denoted by P(E)P(E) and is defined as
P(E)=n(E)/n(S) =number of elements in E/number of elements in S
Let E is the event of occurrence of a head
E = {H}
B). When a die is tossed, sample space S = {1, 2, 3, 4, 5, 6}
Let A is an event of occurrence of an odd number
And B is an event of occurrence of a number greater than 4
A = {1, 3, 5} and B = {5, 6}
P(A)=Probability of occurrence of an odd number=n(A)/n(S)
= 3/6= 1/2
P(B)=Probability of occurrence of a number greater than 4=n(B)/n(S)
=2/6= 1/3
P(E)=n(E)/n(S) =number of elements in E/number of elements in S
Example:
A). When a coin is tossed, then the sample space is S = {H, T}Let E is the event of occurrence of a head
E = {H}
B). When a die is tossed, sample space S = {1, 2, 3, 4, 5, 6}
Let A is an event of occurrence of an odd number
And B is an event of occurrence of a number greater than 4
A = {1, 3, 5} and B = {5, 6}
P(A)=Probability of occurrence of an odd number=n(A)/n(S)
= 3/6= 1/2
P(B)=Probability of occurrence of a number greater than 4=n(B)/n(S)
=2/6= 1/3
Solved examples:
1. A number X is chosen at random from the numbers -3, -2, -1, 0, 1, 2, 3. What is the probability that |X|<2
A. 5/7
A. 5/7
B. 3/7
C. 3/5
D. 1/3
Solution- Option(B) is correct
|X||X| can take 7 values.
To get |X|<2 ( i.e., −2<X<+2) take X={−1,0,1}X={−1,0,1}
To get |X|<2 ( i.e., −2<X<+2) take X={−1,0,1}X={−1,0,1}
⇒ P(|X|<2)= Favorable Cases/Total Cases
= 3/7
2. An urn contains 6 red, 5 blue and 2 green marbles. If 2 marbles are picked at random, what is the probability that both are red?
A. 6/13
B. 5/26
C. 5/13
D. 7/26
solution- Option(B) is correct
P(Both are red),
=6C2/13C2
=5/26
3. Four dice are thrown simultaneously. Find the probability that all of them show the same face.
A. 1/216
B. 1/36
C. 4/216
D. 3/216
Solution- Option(A) is correct
The total number of elementary events associated to the random experiments of throwing four dice simultaneously is:
=6×6×6×6=6^4
n(S)=6^4
Let X be the event that all dice show the same face.
X={(1,1,1,1,),(2,2,2,2),(3,3,3,3),(4,4,4,4),(5,5,5,5),(6,6,6,6)}
n(X)=6
Hence required probability,
=n(X)/n(S)=6/6^4
=1/216
4. A bag contains 12 white and 18 black balls. Two balls are drawn in succession without replacement.
What is the probability that first is white and second is black?
A. 18/145
B. 18/29
C. 36/135
D. 36/145
Solution- Option(D) is correct
The probability that first ball is white:
=12C1/30C1
=12/30 = 2/5
Since, the ball is not replaced; hence the number of balls left in bag is 29.
Hence, the probability the second ball is black:
=18C1/29C1
=18/29
Required probability,
=(2/5)×(18/29)
=36/145
