Basic terminologies –
Experiment – An operation that can produce specific results. E.g – rolling of
dice, tossing of coins.
Random experiment – Any experiment whose outcome cannot be predicted or
determined in advance.
Total number of outcomes (Sample space) – The set of all possible outcomes for
a particular random experiment is called sample space. If N denotes number of
outcomes in a single trail & m denotes the number of times trails or events are
performed then number of possible outcomes = n m
For example,
1.When a dice is thrown once, it results in any one of the following number
(1,2,3,4.5,6)
Sample space(S) = 1,2,3,4,5
2.When a coin is tossed twice,
Sample space (S) = (HH, TT, HT, TH)
● First toss: Heads (H), Second toss: Heads (H) → HH
● First toss: Heads (H), Second toss: Tails (T) → HT
● First toss: Tails (T), Second toss: Heads (H) → TH
● First toss: Tails (T), Second toss: Tails (T) → TT
Example 1 – Two coins are tossed once. Find the sample space.
Answer – Thus, the sample space is
S = {HH, HT, TH, TT} = four possible outcomes.
Example 2 – From a group of two boys and two girls, we select two children.
What should be the sample space for this experiment?
Answer – Here, total number of children = 2 + 2 = 4
4 4 X3
= = 6 ways.
2 X2 2
Example 3 – A coin is tossed initially; if the tail occurs, a ball is drawn from a
box of two red and two white balls, but if the head occurs, we throw a dice. Find
the sample space for this experiment.
Answer – if tail occurs – Red1, Red2, White1, white2
If heads occurs – 1,2,3,4,5,6
Total number of outcomes = 4+6 = 1o outcomes.
Events – A subset of a sample space is known as an event.
For example – Suppose a boy throws a die and reports that the number that
appeared on the die is an even number. In this case, he should have got 2, 4, or
6 because these are the only even numbers in the sample space.
Answer – Sample space (S) = {1, 2, 3, 4, 5, 6}
Event (E) = {2, 4, 6}
Event is a subset of sample space
Occurrence of events –
When an outcome satisfies the condition mentioned in the event completely,
then we can say that the event has occurred.
For example: Suppose we throw a die, and the event (E) = a number less than 3
appears on the die.
Answer – If 1 or 2 appears we say that event has occurred, if 3,4,5,6 occurs on
die, we say event has not occured.
Types of Events –
Simple event – if an event has only one sample point of the sample space, it is
called a simple event.
For Example: Suppose a boy throws a coin.
Sample space (S) = {H, T}
There are four subsets of sample space (S) = {H}, {T}, {HT}, {f}
In this case {H} and {T} are simple events.
Compound event – The events with more than one sample point of the sample
space are known as compound events.
In the previous example {HT} is a compound event.
Equally Likely Events – When a coin is thrown, the probability of head or tail is
1
, we can say that H & T are equally likely events. Similarly, when a dice is
2
1
rolled, probability of getting 1,2,3,4,5,6 is equal to .
6
Then {1} {2} {3} {4} {5} {6} are equally likely events.
Sure Event –
Let S be a sample space associated with a random experiment of rolling a die.
Here S =1,2,3,4,5,6. The set S is also subset of S (S ⊆ S). Since every outcome of
the experiment is a member of S, one can say that S is a certain event.
Impossible event – An event associated with a random experiment is known as
an impossible event if it never occurs whenever experiment is performed.
For example:
Choosing a yellow ball from a bag containing three red balls and five blue balls
is an im-
possible event.
Complement of an Event –
The complement of an event E is the set of
all elements of sample space (S) that are
Complement of an event E is a set of all elements of sample space (S) that are
not in E. They are denoted as E' , E , Ec
Thus,
E′ = S – E
E′ + E = S
If we divide this throughout by S, we can say
E' E S
+ =
S S S
Which gives us P(E′) + P(E) = 1
Now, we can conclude that the sum of the probability of occurrence of an event
and the probability of non-occurrence of an event is equal to 1.
Example – Suppose a person tosses a coin, and E is the event of getting heads
1
P(E) =
2
1 1
Also P¿) = 1 - =
2 2
Now, we can conclude that the complement of getting a head is nothing but
getting a
Tail.
Algebra of Events –
Now, we have understood that a sample space is a set of all possible outcomes
of an experiment, and the events are subsets of the sample space. In this
section, we will learn how new events can be constructed by combining two or
more events.
If we assume that sample space is a universal set for these events, then we have
the following results –
Let A, B, and C be the events of a sample space S.
