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:
Mutually Exclusive EventsBack to Top
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 EventBack to Top
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 EventBack to Top
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 EventsBack to Top
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 EventsBack to Top
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 EventsBack to Top
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 EventsBack to Top
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.