In this article, we will study Probability for Banking and SSC CGL exams from the Quantitative Aptitude section. 

The word probability is very frequently used in our daily life. We usually say, 'He may come today' or 'probably it may rain today' or 'most probably I will clear the exam'. Look at all these phrases and you will realise they contain an element of uncertainity. So to measure these uncertainities, we use the concept of probability. In simple words, probability is a quantitative measure of the certainity. 

The concept of probability has its origins in games of chance, where the outcome of a trial in unknown and uncertain. In the current scenario, concepts of probability have made their way in the world of business and commerce. 

Some important terms to be used while learning probability.

1. Random experiment or trial: The performance of an experiment is called a trial. An experiment is defined by the property that the observations of an experiment under a set of conditions do not always lead to the same observed result, but rather to the different outcomes. If in an experiment, all the possible outcomes are already known and none of the outcomes can be predicted with certainity, then such an experiment is known as Random Experiment. For example, tossing a coin or throwing a die are termed as random experiments. 

2.Event: The possible outcome of an experiment is called an event. The events are depicted with the help of capital letters, A, B, C etc. 

For example: when a coin is tossed the outcome of getting a head or a tail is an event. 

3. Sample space: It is a set of all possible outcomes of an event. It is denoted by letter S. 

For example: when a coin is tossed, S={H,T}, where H = head, T = tail. 

When two coins are tossed, S={HH, TT, HT, TH}

4. Equally likely events: Events are said to be equally likely if there is no preference of any one event to occur in preference to the other. In simple words, equally likely events mean that outcome is as likely to occur as any other outcome. 

For example: when a die is thrown, all the 6 faces {1,2,3,4,5,6} are equally likely to occur. 

5. Exhaustive events: It is the total number of all possible outcomes of any trial. 

For example: when a coin is tossed, either head or tail may turn up and therefore, there are two exhaustive cases. 

there are 6 exhaustive cases or events in throwing a die. 

if 2 dice are thrown simultaneously, there are 36 exhaustive cases. Similarly in case of 3 dice thrown simultaneously, 216 exhaustive events are there. 

6. Algebra of Events: Let us say, A and B are two events with sample space A, then

  1.  is the event that either A or B or both occur.
  2.  is the event that A and B both occur simultaneously.
  3.  is the event that A does not occur. 
  4.  is an event of non-occurence of both A and B, i.e. none of the events A and B occurs. 

For example: In a single throw of a die, let A be the event of getting an odd number and B be the event of getting a number greater than 2. Then, A = {1,3,5} and B = {3,4,5,6}, therefore  = {1,3,4,5,6}. 

 is the event of getting an odd number or a number greater than 2. 

={3,5} is the event of getting an odd number and a number greater than 2. 

 = {2,4,6} is the sample space of elements which are not a part of A or the event of not getting an odd number. 

 = {1,2} is the event of not getting number greater than 2. 

 = {2} is the event of neither getting an odd number nor a number greater than 2. 

7. Mutually Exclusive Events: If in an experiment, an event rules out the happening of the all other events in the same experiment. 

For example: when a coin is tossed either head or tail will appear. Head and tail cannot appear simultaneously in this case. So, occurence of head or tail are called two mutually exclusive events. 

POINT TO REMEMBER: A and B are two mutually exclusive events, which means , i.e. A and B are disjoint sets.

8. Mutually Exclusive and Exhaustive Events: Events  are mutually exclusive and exhaustive if , and  for all 

For example: in a single throw of a die, let A be the event of getting an even number and B be event of getting odd numbers, then A = {2,4,6}, B = {1,3,5},  = {1,2,3,4,5,6} = S. Therefore A and B are mutually and exhaustive events. 

Definition of Probability of an Event: 

It is defined in the following two ways:

  1. Mathematical or a priori definition
  2. Statistical or empirical definition

Mathematical or a priori definition: Probability of an event A, which is denoted as P(A) is defined as, P(A) = (Number of cases favorable to A)/(Number of possible outcomes). 

If an event A happens in 'm' ways and fails, i.e. does not occur in 'n' ways and each of total 'm+n' ways is equally likely to occur then the probability of happening of the event A is given by,  and that the probability of non-occurence of the A is given by, 

Always remember, 

The probability of an event, defined in the above mentioned way is called Priori Probability, i.e. it is determined before hand, i.e. before the actual trials are done. 

Odds of an Event: Let, there are 'm' outcomes favourable to a certain event and 'n' outcomes unfavourable to the event in a sample space, then odds in favour of the event is given as = (Number of favourable outcomes)/(Number of unfavourable events) = m/n. 

Odds against the event = (Number of unfavourable outcomes)/(Number of favourable outcome) = n/m.

Fundamental theorems on Probability:

Theorem 1:  For a random experiment, let S is the sample space and E is an event, then

  1. probability of occurence of an event is always non-negative, .
  2. probability of occurence of an impossible event is 0, .
  3. probability of occurence of a sure event is 1.

Theorem 2: Let A and B are two mutually exclusive events, then

  1. .

Theorem 3: Let A and B are two mutually exclusive and exhaustive events them 

Theorem 4: Let A be any event and  be its complimentary event, then .

Theorem 5: For any two events A and B, 

Theorem 6: Addition theorem - For any two events A and B, 

If A and B are mutually exclusive events, then .

Theorem 7: For three events A, B and C

Example: A card is drawn at random from a well-shuffled pack of 52 cards. Find the probability of getting, (i) a jack or a queen or a king, (ii) a two of heart or diamond.

Solution: (i) In a pack of 52 cards, we have 4 jacks, 4 queens and 4 kings. So, a jack and a queen and a king are mutually exclusive events. 

P(a jack) = 4/52 = 1/13

P(a queen) = 4/52 = 1/13

P(a king) = 4/52 = 1/13

By addition theorem of probability, 

P(a jack or a queen or a king) = P(a jack) + P(a queen) + P(a king) = 1/13 + 1/13 + 1/13 = 3/13.

(ii) P(two of heart or two of diamond) = P(two of heart) + P(two of diamond) = 1/52 + 1/52 = 2/52 = 1/26.

Independent events: Two events A and B are called independent events if the occurence or non-occurence of one does not affect the the probability of occurence of the other. 

For example: in a simultaneous throwing of 2 coins, getting a tail on the first con and getting a head on the second coin are two independent events. 

 

Comments

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Thanks for this Sir. Can you share some practice questions also? TIA

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