Sample spaces, events and types of events, Algebra of events:

The set of all possible outcomes of a random experiment is called the sample space associated with the experiment.

Sample space is denoted by the symbol S.

Each element of the sample space is called a sample point.

 In other words, each outcome of the random experiment is also called sample point.

Example : Two coins (a one rupee coin and a two rupee coin) are tossed once.

Find a sample space.

Solution:

Heads on both coins = (H,H) = HH

Head on first coin and Tail on the other = (H,T) = HT

Tail on first coin and Head on the other = (T,H) = TH

Tail on both coins = (T,T) = TT

Thus, the sample space is S = {HH, HT, TH, TT}

Example : Consider the experiment in which a coin is tossed repeatedly until a head comes up. Describe the sample space.

Solution:

In the experiment

head may come up on the first toss, or the 2nd toss, or the 3rd toss and so on till head is obtained.

Hence, the desired sample space is  S= {H, TH, TTH, TTTH, TTTTH,...}

Event:

Any subset E of a sample space S is called an event.

Types of events:

  1. Impossible and Sure Events :

The empty set f and the sample space S describe events.

In fact f is called an impossible event and S, i.e., the whole sample space is called the sure event.

Example:

let us consider the experiment of rolling a die. The associated sample space is

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

Let E be the event “ the number appears on the die is a multiple of 7”.

E={} = f is an impossible event.

Let event F “the number turns up is odd or even”.

F= {1, 2, 3, 4, 5, 6} = S

Thus, the event F = S is a sure event.

  1. Simple Event : If an event E has only one sample point of a sample space, it is

called a simple (or elementary) event.

In a sample space containing n distinct elements, there are exactly n simple events.

For example in the experiment of tossing two coins, a sample space is

S={HH, HT, TH, TT}

There are four simple events corresponding to this sample space. These are

E1= {HH}, E2={HT}, E3= { TH} and E4={TT}.

  1. Compound Event: If an event has more than one sample point, it is called a

Compound event.

For example, in the experiment of “tossing a coin thrice” the events

E: ‘Exactly one head appeared’

F: ‘Atleast one head appeared’

G: ‘Atmost one head appeared’ etc.

are all compound events. The subsets of S associated with these events are

E={HTT,THT,TTH}

F={HTT,THT, TTH, HHT, HTH, THH, HHH}

G= {TTT, THT, HTT, TTH}

Algebra of events:

  1. Complementary Event
  2. The Event ‘A or B’
  3. The Event ‘A and B’
  4. The Event ‘A but not B’

i)  Complementary Event: For every event A, there corresponds another event Acalled the complementary event to A. It is also called the event ‘not A’.

For example, take the experiment ‘of tossing three coins’. An associated sample

Space  is S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}

Let  A= {HTH, HHT, THH} be the event ‘only one tail appears’

Clearly for the outcome HTT, the event A has not occurred. But we may say that the event ‘not A’ has occurred. Thus, with every outcome which is not in A, we say that ‘not A’ occurs.

Thus the complementary event ‘not A’ to the event A is

A`= {HHH, HTT, THT, TTH, TTT}

or  A¢ = {w : w Î S and w ÏA} = S – A.

ii)  The Event “ A or B  “ : Recall that union of two sets A and B denoted by A È B contains all those elements which are either in A or in B or in both.

When the sets A and B are two events associated with a sample space,

Then ‘A È B’ is the event ‘either A or B or both’. This event ‘A È B’ is also called ‘A or B’.

Therefore Event ‘ A or B ’ = A È B  = {w : w Î A or w Î B}

iii)  The Event ‘ A and B ’ : We know that intersection of two sets A Ç B is the set of

those elements which are common to both A and B. i.e., which belong to both ‘A and B’.

If A and B are two events, then the set A Ç B denotes the event ‘A and B’.

Thus, A Ç B = {w : w Î A and w Î B}

For example, in the experiment of ‘throwing a die twice’

Let A be the event ‘score on the first throw is six’ and

 B is the event ‘sum of two scores is atleast 11’ then

A = {(6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}, and B = {(5,6), (6,5), (6,6)}

So  A Ç B = {(6,5), (6,6)}

Note that the set A Ç B = {(6,5), (6,6)} may represent the event ‘the score on the first throw is six and the sum of the scores is atleast 11’.

iii)  The Event ‘A but not B’ : We know that A–B is the set of all those elements

which are in A but not in B. Therefore, the set A–B may denote the event ‘A but not B ’.We know that  A – B = A Ç

Example:  Consider the experiment of rolling a die. Let A be the event ‘getting a prime number’, B be the event ‘getting an odd number’. Write the sets representing the events

(i) Aor B (ii) A and B (iii) A but not B (iv) ‘not A’.

Solution:  Here S = {1, 2, 3, 4, 5, 6}, A = {2, 3, 5} and B = {1, 3, 5}

Obviously

(i) ‘A or B’ = A È B = {1, 2, 3, 5}

(ii) ‘A and B’ = A Ç B = {3,5}

(iii) ‘A but not B’ = A – B = {2}

(iv) ‘not A’ = A¢ = {1,4,6}

Mutually exclusive events:

Two events A and B are called mutually exclusive events if the occurrence of any one of them excludes the occurrence of the other event, i.e., if they can not occur simultaneously. In this case the sets A and B are disjoint.

Example:

In the experiment of rolling a die, a sample space is

S = {1, 2, 3, 4, 5, 6}. Consider events, A ‘an odd number appears’ and B ‘an even

number appears’

Clearly the event A excludes the event B and vice versa.

In other words, there is no outcome which ensures the occurrence of events A and B simultaneously.

Here A = {1, 3, 5} and B = {2, 4, 6}

Clearly A Ç B = f, i.e., A and B are disjoint sets.

So two events A and B are called mutually exclusive events.

Note: Simple events of a sample space are always mutually exclusive.

if E1, E2, ..., En are n events of a sample space S and ifE1 È E2 È E3 È E4 È...... È En = S

then E1, E2, ...., En are called exhaustive events.

In other words, events E1, E2, ..., En are said to be exhaustive if atleast one of them necessarily occurs whenever the experiment is performed.

Further, if Ei Ç Ej = f for i ¹ j i.e., events Ei and Ej are pairwise disjoint and

, then events E1, E2, ..., En are called mutually exclusive and exhaustive events.

Example :  A coin is tossed three times, consider the following events.

A: ‘No head appears’, B: ‘Exactly one head appears’ and C: ‘Atleast two heads

appear’.

Do they form a set of mutually exclusive and exhaustive events?

Solution: The sample space of the experiment is

S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}

and A = {TTT}, B = {HTT, THT, TTH}, C = {HHT, HTH, THH, HHH}

Now

A È B È C = {TTT, HTT, THT, TTH, HHT, HTH, THH, HHH} = S

Therefore, A, B and C are exhaustive events.

Also, A Ç B = f, A Ç C = f and B Ç C = f

Therefore, the events are pair-wise disjoint, i.e., they are mutually exclusive.

Hence, A, B and C form a set of mutually exclusive and exhaustive events.