Mathematics PROBABILITY- Event , Types of events
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### Topics Covered

star Event
star Occurrence of an event
star Impossible and Sure Events
star Simple Event
star Compound Event
star Complementary Event
star The Event ‘A or B’
star The Event ‘A and B’
star The Event ‘A but not B’
star Mutually exclusive events
star Exhaustive events

### Event

\color{fuchsia} {ul ★ "Event "}

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

Consider the experiment of tossing a coin two times. An associated sample space is color(navy)(S = {HH, HT, TH, T T}.)

Now suppose that we are interested in those outcomes which correspond to the occurrence of exactly one head.

We find that HT and TH  are the only elements of S corresponding to the occurrence of this happening (event). These two elements form the set color(navy)(E = { HT, TH})

We know that the set E is a subset of the sample space S . Similarly, we find the following correspondence between events and subsets of S.

color(red)"Description of events "  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \color(red) " Corresponding subset of ‘S’"
Number of tails is exactly 2  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ A = {T T}
Number of tails is atleast one  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ B = {HT, TH, T T }
Number of heads is atmost one  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ C = {HT, TH, T T}
Second toss is not head  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ D = { HT, T T}
Number of tails is atmost two  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ S = {HH, HT, TH, T T}
Number of tails is more than two \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ φ

### Occurrence of an event

\color{red} ✍️  color(blue)("The event E of a sample space S is said to have occurred") if the outcome ω of the experiment is such that color(blue)(ω ∈ E.)

\color{red} ✍️  If the outcome ω is such that color(blue)(ω ∉ E), we say that color(blue)("the event E has not occurred.")

color(red)("Consider the experiment of throwing a die.")

Let E denotes the event color(blue)"“ a number less than 4 appears”."

If actually ‘1’ had appeared on the die then color(green)("we say that event E has occurred.")

As a matter of fact if outcomes are 2 or 3, we say that event E has occurred .

### Impossible and Sure Events

The empty set φ and the sample space S  describe events. In fact φ is color(red)("called an impossible event") and

color(blue)("S, i.e., the whole sample space is called the sure event.")

color(red)(=>"To understand these let us consider the experiment of rolling a die. ")

The associated sample space is color(blue)(S = {1, 2, 3, 4, 5, 6})

Let E be the event color(green)"“ the number appears on the die is a multiple of 7”".

Clearly no outcome satisfies the condition given in the event, i.e., no element of the sample space ensures the occurrence of the event E.

Thus, we say that the empty set only correspond to the event E.

In other words we can say that it is impossible to have a multiple of 7 on the upper face of the die.

Thus, the event E = φ is an color(blue)"impossible event."

color(red)(=>"Now let us take up another event F") color(blue)"“the number turns up is odd or even”."

Clearly color(blue)(F = {1, 2, 3, 4, 5, 6,} = S), i.e., all outcomes of the experiment ensure the occurrence of the event F.

Thus, the event color(red)(F = S) is a color(red)"sure event."

### Simple Event

\color{fuchsia} {ul ★ "Simple Event "}

If an event E has only one sample point of a sample space, it is called a color(blue)"simple (or elementary) event."

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

color(red)(=>"For example") in the experiment of tossing two coins, a sample space is

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \S={HH, HT, TH, T T}

There are color(blue)("four simple events") corresponding to this sample space. These are

E_1= {HH}, \ \ \ \ \ E_2={HT}, \ \ \ \ \ \ E_3= { TH} and E_ 4={T T}.

### Compound Event

\color{fuchsia} {ul ★ "Compound Event "}

If an event has more than one sample point, it is called a color(blue)"Compound event."

color(red)(=>"For example"), in the experiment of color(navy)"“tossing a coin thrice”" the events

" " color(red)(E : ) color(blue)("‘Exactly one head appeared’")

" " color(red)(F : ) color(blue)("‘Atleast one head appeared’")

" " color(red)(G : ) color(blue)("‘Atmost one head appeared’") etc.

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

color(red)(E )={HT T,THT,T TH}

color(red)(F )={HT T,THT, T TH, HHT, HTH, THH, H H H}

color(red)(G)= {T T T, THT, HT T, T TH}

Each of the above subsets contain more than one sample point, hence they are all "compound events."

### Complementary Event

\color{fuchsia} {ul ★ "Complementary Event "}

\color{red} ✍️ For every event A, there corresponds another event A′ called the complementary event to A.

\color{red} ✍️ It is also called the event color(blue)"not A".

\color{red} ✍️ The complementary event color(blue)("‘not A’") to the event A is

or color(green)(A′ = {ω : ω ∈ S and ω ∉A} = S – A.)

color(red)(=>"For example,") take the experiment color(blue)("‘of tossing three coins’.")

An associated sample space is

\ \ \ \ \ \ color(blue)(S = {HHH, HHT, HTH, THH, HT T , THT, T T H, T T T})

Let color(green)(A={HTH, HHT, THH}) be the event color(green)"‘only one tail appears’"

Clearly for the outcome HT T, 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 color(blue)("‘not A’") to the event A is

A′ = {HHH, HT T, THT, T TH, T T T}

### The Event ‘A or B’

\color{red} ✍️ Recall that union of two sets A and B denoted by A ∪ B contains all those elements which are either in A or in B or in both.

\color{red} ✍️ When the sets A and B are two events associated with a sample space, then A ∪ B is the event color(red)"‘either A or B or both’."

\color{red} ✍️ This event color(red)(A ∪ B) is also called color(red)"A or B".

\color{red} ✍️ Therefore Event color(green)("A or B"= A ∪ B = {ω : ω ∈ A or ω ∈ B})

### The Event ‘A and B’

\color{red} ✍️ We know that intersection of two sets A ∩ B is the set of those elements which are common to both A and B. i.e., which belong to both color(red)"‘A and B’."

\color{red} ✍️ If A and B are two events, then the set color(red)(A ∩ B) denotes the event color(red)"‘A and B’."

\color{red} ✍️ Thus, color(green)(A ∩ B = {ω : ω ∈ A and ω ∈ B})

### The Event ‘A but not B’

\color{red} ✍️ We know that A–B is the set of all those elements which are in color(blue)"A but not in B."

\color{red} ✍️ Therefore, the set A–B may denote the event "‘A but not B’ " .

\color{red} ✍️ We know that color(blue)(A – B = A ∩ B´)
Q 3120091811

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) A or 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 ∪ B = {1, 2, 3, 5}
(ii) ‘A and B’ = A ∩ B = {3,5}
(iii) ‘A but not B’ = A – B = {2}
(iv) ‘not A’ = A′ = {1,4,6}

### Mutually exclusive events

\color{fuchsia} {ul ★ "Mutually exclusive events "}

\color{red} ✍️ Two events A and B are called color(green)"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.

\color{red} ✍️ color(blue)("In this case the sets A and B are disjoint.")

color(red)(=>" In the experiment of rolling a die"), a sample space is S = {1, 2, 3, 4, 5, 6}.

Consider events, A color(blue)"‘an odd number appears’" and B color(blue)"‘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 color(blue)(A = {1, 3, 5}) and color(blue)(B = {2, 4, 6})

Clearly color(blue)(A ∩ B = φ,) i.e., A and B are disjoint sets.

color(red)(=>"Again in the experiment of rolling a die")

Consider the events color(blue)("A ‘an odd number appears’") and event color(blue)("B ‘a number less than 4 appears’")

Obviously A = {1, 3, 5} and B = {1, 2, 3}

Now 3 ∈ A as well as 3 ∈ B

Therefore, A and B are color(blue)"not mutually exclusive events."

### Exhaustive events

\color{fuchsia} {ul ★ "Exhaustive events "}

Such events A, B  and C are color(green)("called exhaustive events.")

In general, if E_1, E_2, ..., E_n are n events of a sample space S and if

color(blue)(E_1 uu E_2 uu E_3 uu ............. uu E_n = underset(i = 1) overset(n) uu E_i = S)

then E_1, E_2, ...., E_n are called exhaustive events.

In other words, events E_1, E_2, ..., E_n are said to be exhaustive if atleast one of them necessarily occurs whenever the experiment is performed.

color(red)(=>"Consider the experiment of throwing a die.")

We have S = {1, 2, 3, 4, 5, 6}. Let us define the following events

" " color(blue)"A : ‘a number less than 4 appears’,"

" "color(blue)" B : ‘a number greater than 2 but less than 5 appears’" and

" " color(blue)"C : ‘a number greater than 4 appears’."

Then A = {1, 2, 3}, B = {3,4} and C = {5, 6}.

We observe that color(blue)(A uu B uu C = {1, 2, 3} uu {3, 4} uu {5, 6} = S.)

\color{fuchsia} {ul ★ "Mutually exclusive and Exhaustive events "}

\color{red} ✍️ Further, if color(blue)(E_i ∩ E_j = φ) for color(red)(i ≠ j) i.e., events E_i and E_j are pairwise disjoint and

underset(i = 1) overset(n) uu E_i = S then events E_1 , E_2 , ......... E_n are called mutually exclusive and exhaustive events.
Q 3140091813

Two dice are thrown and the sum of the numbers which come up on the dice is noted. Let us consider the following events associated with this experiment
A: ‘the sum is even’.
B: ‘the sum is a multiple of 3’.
C: ‘the sum is less than 4’.
D: ‘the sum is greater than 11’.

Which pairs of these events are mutually exclusive

Solution:

There are 36 elements in the sample space S = {(x, y): x, y = 1, 2, 3, 4, 5, 6}.
Then
A = {(1, 1), (1, 3), (1, 5), (2, 2), (2, 4), (2, 6), (3, 1), (3, 3), (3, 5), (4, 2), (4, 4),
(4, 6), (5, 1), (5, 3), (5, 5), (6, 2), (6, 4), (6, 6)}
B = {(1, 2), (2, 1), (1, 5), (5, 1), (3, 3), (2, 4), (4, 2), (3, 6), (6, 3), (4, 5), (5, 4), (6, 6)}
C = {(1, 1), (2, 1), (1, 2)} and D = {(6, 6)}
We find that
A ∩ B = {(1, 5), (2, 4), (3, 3), (4, 2), (5, 1), (6, 6)} ≠ phi
Therefore, A and B are not mutually exclusive events.
Similarly A ∩ C ≠ phi, A ∩ D ≠ phi, B ∩ C ≠ phi "and" B ∩ D ≠ phi.
Thus, the pairs, (A, C), (A, D), (B, C), (B, D) are not mutually exclusive events.
Also C ∩ D = phi and so C and D are mutually exclusive events.