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Concept of Probability – Part 1

Concept of Probability – Part 1

We often use the term “probability” in many situations without understanding a definite meaning of it. For example, like we say ‘the probability that it will rain today is 30%’, ‘the probability of getting a head in tossing a coin is 50%’ etc. First of all, we need to understand what does probability actually mean. The word ‘probability’ means ‘chance’, as simple as that. But that chance is based on certain experiments, rather we say ‘random experiments’.

 

Now we know that if we are enthusiastic enough, we can do a little bit of experiments with tossing a coin or rolling a dice. What we get as the results from these experiments are called ‘outcome’. And when we do any such experiments, we should be honest and walk down the street of truth.

 

What is a Random Experiment?

An experiment is called random experiments if

  1. if we know all the outcomes or results that may occur if we perform the experiment,
  2. though we know all the outcomes that can happen, it will be impossible for us to predict,
  3. the experiment can be repeated a number of times under the same or identical situations.

Example: Suppose if we roll a dice, it is an example of random experiment.

  1. Now we know that we are going to get from 1 to 6.
  2. But We can not predict that we will surely get 4 or 6.
  3. Yes we can repeat the rolling of dice whole day if we are not bored J

Note: An Experiment is denoted by E.

Now thus far you have understood the very basic definition of “probability”. What is it? It is we can not predict what will happen exactly, but we have a set of options and the result of an experiment will come from that set and our job is to calculate the chance of getting any outcome from that set which may occur.

Let us learn some definitions with examples:

  1. Event Space:

The set of all possible outcomes of a random experiment is called event space.

Example: Suppose in a random experiment of rolling a die, we know that we are going to from 1 to 6. Now the purpose or condition is simple here just roll a dice and get the outcomes. If we denote the event space by S, then S = {1, 2, 3, 4, 5, 6}. Similarly, for tossing a coin, event space will be S = {H, T}, that is H = head, T = tail.

 

  1. Event:

An event A of a random experiment can be defined as a subset of the corresponding event space S. Well, we will give here one example to simplify the definition.

 

Example: Again we are rolling a die. So what is the event space here? It is denoted by S, and which is S = {1, 2, 3, 4, 5, 6}. Now if we say, “even number appears while throwing a die”, that is called an event, its set of outcomes is denoted by A, which is A = {2, 4, 6}. Now clearly A is the subset of S.

 

  1. Composite and Compound Event:

If an event can be composed into events, then that event is called composite or compound events.

Example: Let A and B be the events ‘even face’ and ‘odd face’ respectively connected with random experiment of rolling an unbiased dice. Now, we can say A has happened if we get ‘two’, ‘four’, ‘six’, and again we can say B has happened if we get ‘one’, ‘three’, ‘five’. Thus events A and B have been composed into events, such events like A and B are called composite and compound events.

 

  1. Simple Events:

If we cannot decompose an event into events, then that event is called Simple Event.

Example: In the above example, we can not decompose events ‘one’, ‘two’, ‘three’ etc any further. Then these events are called Simple Events.

 

  1. Impossible Events:

An event connected with a random experiment is called Impossible Event if we think of an event that is logically impossible to happen in any conditions.

Example: Getting ‘seven’ in throwing a dice is an impossible event. An impossible event is denoted by O where clearly O is an empty set.

 

  1. Certain Events:

In any random experiment if we think of an event which is sure to happen in any performance of the experiment, then that event is called Certain Event.

Example: Let A denote the event ‘Head or Tail’ connected with a random experiment of tossing a coin, then this event is going to happen in every toss. We call this event as Certain Event. A certain event is denoted by S.

 

  1. Mutually Exclusive Events:

Two events connected with a random experiment with E are called mutually exclusive events if they cannot occur simultaneously. Symbolically, if two events A and B are mutually exclusive events when A ∩ B = Φ, where Φ is the impossible event.

Example: Let A denotes the event ‘even number occurs’ and B denotes the event ‘odd number occurs’ connected with a random experiment of throwing a dice. Then A and B become mutually exclusive events since in any performance of the experiment, both A and B cannot occur simultaneously.

Again, let X denotes ‘multiple of three’ and Y denotes ‘Even number’ connected with a random experiment of throwing a dice. In that case, if we get 6, then we see that both X and Y occur simultaneously. So X and Y are not mutually exclusive events here.

 

  1. Complementary Events:

Let A be an event connected with a random experiment E. The event of non occurance of A is called the complementary of to A and is denoted by Ac or A/ or .

Example: Suppose we take a random experiment of drawing a card from a well shuffled pack of 52 cards and the event A is of drawing a spade from the pack, now the complementary to A is the event of not drawing a spade from the same pack.

 

  1. Equally Likely Events:

Suppose we are having 2 or more events connected with a random experiment E. Now if we are not able to give preference to one another after taking into consideration all relevant evidence, then we will call those events equally likely events.

Note: Symbolically if A and B are equally likely events then P(A) = P(B).

Example: Simple events are equally likely events but compound events may or may not be so. If we consider a random experiment of tossing a coin, now ‘HEAD’ and ‘TAIL’ are connected with it and they are simple events and obviously they are equally likely events. We cannot provide a preference to one another.

The compound events ‘EVEN FACE’ and ‘MULTIPLE OF THREE’ are not equally likely events but compound events ‘ODD FACE’ and ‘EVEN FACE’ are so.

 

  1. Exhaustive Events:

A collection of events connected with a random experiment is said to be Exhaustive Events if in every performance of the experiment at least one event, not necessarily the same for every performance, belonging to the collection happens.

Simple events connected with the random experiments are Exhaustive Events.

Example: Suppose if we consider a random experiment of throwing a die, we are having simple events ‘ONE’, ‘TWO’, ‘THREE’, ‘FOUR’, ‘FIVE’, ‘SIX’. Clearly at least one of them is sure to happen and they form a exhaustive set of events.

Now suppose we are having three events A, B, C which are ‘EVEN FACE’, ‘ONE’, ‘MULTIPLE OF THREE’ respectively connected with the throwing a die random experiment. Now if we get ‘FIVE’ surely we can say that none of them has occurred. Then A, B, C are not exhaustive events.

 

Avijit Nandi
Alumni of Jadavpur University

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