`\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`

`\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 .

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."`

`\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}.`

`\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."`

`\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}`

`\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})`

`\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})`

`\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´)`

`\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."`

`\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.