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Probability of Occurrence of an Event

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Let us consider any event E, which is associated to some of the random experiment and is occur if the occurrence of any one of the elementary event is its outcome. If S is a Sample Space and a random experiment is performed. Then if E is any event, it means E is the subset of S. Then the Probability of an occurrence of event E is defined as:
P (E) = number of distinct elements of Event E / Number of distinct elements of sample S. We also know that S includes all possible outcomes of the event.
Or we can write P (E) = n (E) / n (S)
Let us take some example to understand the concept more clearly. Let’s take a pot with 10 blue balls and 90 red balls, which are of similar shape and size. Let us first mix the balls thoroughly and then one ball is drawn from the pot randomly. We come to the conclusion that the ball taken out from the pot will be either of blue color or of red color. As the number of red balls is much more than the number of blue balls, so this is more likely to get a Red Ball in compare to the Blue Ball. So we can say that the probability of getting a red ball is more likely than the probability of getting a blue color ball. This means event which is more likely to occur has higher probability as compared to the event which has less possibility of occurrence. Here we first count the total number of events = 90 + 10 = 100.  We say that the probability of an occurrence of an event of getting a red ball = P( Red Ball ) = 90 / 100. And the probability of an occurrence of an event of getting a blue ball = P ( Blue Ball) = 10 /100.

Mutually Exclusive Events

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Mutually exclusive events are also called as incomparable events. Now let’s look at mutually exclusive events definition: Two or more events are called mutually exclusive events in such a case when we find that the occurrence of one of the event prevents the occurrence of the other events in the space. It also means that no two or more events can occur simultaneously in the same trial. If we look at the Elementary Events related to the random experiments, then they are always mutually exclusive events because these are the results of the experiments, when performed, we get only one outcome at one time. Let us take a dice is thrown at random. Now, Let us take an example of 3 events say X, Y and Z, they represent the following experiments: 
X = Getting an Even Number
Y = Getting an odd number
Z = getting a multiple of a number 3
We observe that X = 2, 4, 6
                          Y = 1, 3, 5 and 
                           Z = 3, 6
 We come to a conclusion X ∩ y = ∅. And x ∩ Z ≠ ∅
 Y ∩ Z ≠ ∅ and x ∩   y ∩ Z = ∅
 
When we say that X ∩ y = ∅, it means that the elements of Set X and set Y have no element in common. On the other hand when we write that x ∩ Z ≠ ∅, it means that their exist at least one element common between X and Z set. Also saying that x ∩   y ∩ Z = ∅ represents that there is no element common in all the three Sets X, Y and Z.
Such events are called mutually exclusive events, which do not have any common element in it. 

Sure Event

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Let us consider n Elementary Events which are associated with any random experiment. Then their exist 2^n Subsets. Each subset of S is an event associated to the random experiment and the given Sample Space is the universal Set of these events. These 2^n events are divided into different types on the basis of their nature of occurrence. When we talk about a sure event, which is also called certain event. An event will be called a sure event or certain event if it always occurs whenever any experiment is performed.
A sure event will be clearer with the help of an example:  Let us take an example of throwing a die. The possible outcomes of the sample space will be 1, 2, 3, 4, 5, 6. Now we consider an event of “Getting an even or odd number” in any particular event. We find that this event is represented with the following elements in the set 1, 2, 3, 4, 5, 6 which is exactly same as the sample space. So it is called sure event. This example can also be seen in case of picking a card from the pack. Let the event be to select black or red card, in this event the Probability is again 100%. So it is a sure event.
This type of event has 100% probability of getting success of the event. It is the inverse of Impossible Event, which indicates that impossible event is the event where there is no outcome of the event and there is 0% probability of getting an outcome of impossible event. 
We can also say P (Sure Event) = 1. It can also be written as P (S) = 1, where S is any sample space. It shows that on every experiment of the event outcome is sure and certain.

Impossible Event

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Every subset of a Sample Space is called an event in language of Probability and Statistics. To every event which is associated with any random experiment, we try to attach any numerical value which is called the Probability. Any event associated with a random experiment is called an impossible event when we observe that this event does not occur at all when ever and how many times the experiment is performed. Let us take any event of rolling a dice. We know that the possible outcomes of rolling a dice experiment will be the Set of   1, 2 , 3 , 4 , 5 , 6 Now in this experiment, if  E is any event " of getting a number greater than or equal to 7" . We find that no number greater than 6 exists in this experiment. So this event is impossible event. We can also say that if S is the sample space with any random experiment, then an empty set call Null set is a subset of the set S which has no outcome so the set φ called as an empty set is used to represent an impossible set in the space S. 
Such events are called impossible events where there is no output and thus we find the output set as an empty set. This impossible set does not include any element from the set of elements in the space and thus its order is 0. We can also say that impossible event is the reverse of Sure Event. We can also say that probability of impossible set is 0. If we have a pack of 52 cards and we see its elements are all 52 cards of red and black color. If we need to find the probability of a blue card, then it is impossible.  As we know that there is no card with blue color. So this event is not possible whenever we pick any card from the pack of 52 cards. Thus we say this event is impossible.

Complementary Events

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Any two events are called complementary if the event has only two possible outcomes. Let us consider a situation of throwing a coin. There are only two possible outcomes in this event, either we get a head or we get a tail. In such event we observe that at one time only one of the two events occur and besides these two events, no other event will occur. Such two events are called complementary events.
Let us take another example of rolling a dice and then to check whether the result is “an even number “or “an odd number”. We conclude that rolling dice and the event of getting “even” or “odd”  is a complementary event as we find that there are only two possible outcomes of rolling a dice, either we get an even number or an odd number on the dice. Moreover we also observe that there is no other possibility of getting any other result than even or odd number. So we say that these two events are also complementary.
Let us say that we have 5 red balls and 3 white balls, in all we have 8 balls. If we take an event of getting red or white ball then this event is also a complement event, as either a ball taken out from bag will be red or it will be white.
 All these are the examples of complementary events in Probability. If any event is represented by A, then the complement of the event is represented by A’.
 In order to find the probability of any two events we observe that if A is any event and A’ is the complement of the event, then A + A’ = 1.
In order to find the probability of any event not occurring, i.e. to find the complement of any event we simply need to subtract the probability of that event from 1.
Thus we conclude that A’ = 1 – A.

Exhaustive Events

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Any two or more events which occur in the space S and are associated with some random experiment are called exhaustive event if their Union forms a Sample Space. Exhaustive events Probability of the union of all the events is always 1. One case of a collectively exhaustive and mutually exclusive event is tossing a coin, whose likely results are Head or Tail. We observe that the probability of getting either a head or getting either a tail will be a complete event. So, P(Head or Tail) = 1, so the outcomes are collectively exhaustive. When head occurs tail can't occur or P(Head and Tail) = 0, so the outcomes are mutually exclusive also, which means either of the one event at a time  can occur at a time. If we take E1, E2, E3, ..... En as the events of the sample space associated with the random experiments, then the space S is an Exhaustive event if the union of all the events.

 E1, E2, E3, ......En will result to the space S            
 It will be clearer with the following example: If a die is thrown and 
If E1 = Getting a even number = 2 , 4 , 6
   E2 = Getting an odd number = 1, 3, 5
In this space we find union of E1 and E2 will result to all the possible elements = 1, 2, 3, 4, 5, 6
 So we can say union of E1 and E2 = S, so we say that the events E1 and E2 are exhaustive events.

Similarly if we take a pack of 52 cards, and we say that the two events be such that 
 E1 = Probability of getting a Black card
 E2 = Probability of getting a Red card
Here again we find that union of E1 and E2 = S; which means the union of two events will include all the 52 cards of the pack.

Elementary Events

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An example of an elementary event is to get the number 4 on the roll of a dice. As the word tells, elementary means only one. Thus, elementary event means the event which has only one outcome. This event is also termed as Atomic event. Any simplest type of event which we can ever talk about is an “elementary event" .It is the most basic type of event which we can talk about. It is also called the smallest event and the simplest event which can ever exist. Elementary events cannot be further decomposed into smaller units. In order to find out the Probability of an elementary event, we can simply say that it is the probability of getting the only outcome contained within the Set.
If we study about the Probability Theory, an elementary event or an atomic event is the event which has only single outcome in the sample given space. It can also be given a name simple event. This directly represents that the event is singleton, which means it has only one element in it. 
All Elementary events and their equivalent outcomes can also be marked identically for simplicity; as such an event relates to exactly one outcome.
If we take an example of throwing a coin, then the possible outcomes of the set will be H, T . Now the two elementary Sets representing the possible outcomes are H, T.  In case the random experiment is performed, then each outcome of the experiment is called an independent event or elementary event. In other words we can also say that outcomes of random experiments are also called elementary events associated to it. So we can say that if E1, E2, E3.... E6 are 6 elementary events associated to any experiment then the space S associated to this experiment will be s = E1, E2, E3,…, E6.

Favourable Elementary Events

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The events which cannot be further divided to any smaller events are called Elementary Events or we can call it simple events
Let us consider any experiment ‘S’ of tossing a coin. We know that the possible outcomes of throwing a coin  S = H, T.
 In this expression, we observe that “ S “ is a Sample Space and “ H “ and “ T “ are the two elementary events of getting a head or a tail.
 We can say that there are certain small events which we cannot break further into smaller ones; these events are called elementary events. We also call it simple events.
If we define favorable event, we say that the event which tell the success of the event which has occurred-
We use the English alphabet "m" to define the success of a certain occurrence of the event. Thus we say 'm' describe the favorable conditions of the event. Favorable condition of the event represents how many times does the success of the event has taken place. It will become clearer with the help of example given below.
 Now let us see what is Favorable Elementary Event? Any elementary (unit) event which gives a result in ones favors is called a favorable independent event .
Let us consider an experiment of tossing a coin, there are two possible results or outcomes i.e. there are two possible favorable elementary events or choices.
The event of getting a HEAD,
The event of getting a TAIL,
 The Probability of getting an elementary event H = ½,
The probability of getting an elementary event T = ½.
Let us consider another experiment of rolling a dice, there are six possible results or outcomes i.e. there are six possible favorable elementary events or choices. They are 1, 2, 3, 4, 5, 6.
In this if we find the probability of getting elementary event 1 = 1/6.