CAT Venn Diagram for probability is the mix of two topics i.e. Venn Diagram and Probability. CAT Question Paper will have questions based on Venn diagrams for probability which can be solved if the candidate has the knowledge of both concepts. Based on CAT Paper Analysis of the previous year’s, the questions are usually of easy to moderate difficulty level. 

In CAT 2021, candidates can expect around 2 or 3 questions based on this topic. Another name of the Venn Diagram is the Euler-Venn diagram. It is represented as sets of diagrams. In the Venn Diagram, the rectangular shape is defined as universal whereas the circle is placed under the rectangular shape. In the below section, a picture of the Venn Diagram is portrayed, 

With the help of figure, it can be concluded that: 

• n(A) = x + z 
• n(B) = y + z 
• n(A∩B) = z
• n(A∪B) = x +y+ z
• Total number of elements = x + y + z + w
• n is the number of elements in a set

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CAT Venn Diagram for Probability: Formulas

Some basic formulas along with the shortcut tips for CAT Venn Diagram for Probability are mentioned below.

• n ( A ∪ B) = n(A ) + n ( B ) - n ( A∩ B)
• n (A ∪ B ∪ C) = n(A ) + n ( B ) + n (C) - n ( A ∩ B) - n ( B ∩ C) - n ( C ∩ A) + n (A ∩ B ∩ C )

In a statement, where n( A) is equal to the number of elements in set A. After understanding the basic concept of a Venn Diagram, candidates can also skip the formulas.

CAT Preparation: Fundamentals of Probability

The basic concepts of Probability are as follows: 

• Probability theory tells us how to work with the probability function and derive ‘probabilities of events’ from it.
• We define probabilities to span the unit interval, i.e. [0, 1], such that for any event A we have:
  • 0 ≤ P (A) ≤ 1.
• Experiment: For example, rolling a single die and recording the outcome.
• Outcome of the experiment: For example, rolling a 3.
• Sample space S: The set of all possible outcomes, here {5,6,7,8}.
• Event: Any subset A of the sample space, for example A = {6, 7, 8}.
• A set is a collection of elements (also known as ‘members’ of the set)

Example: The following are all examples of sets, where ‘|’ can be read as ‘such that’:

1. = {Raj, Suresh, Jitesh}
2. = {1, 2, 3, 4, 5}
3. = {x | x is a prime number} = {2, 3, 5, 7, 11, . . . }
4. = {x | x ≥ 0} (that is, the set of all non-negative real numbers).

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CAT Preparation: Simple Probability Distributions 

• Probability of an event is denoted by P(A)
• One can view probability as a quantifiable measure of one’s degree of belief in a particular event, or set, of interest.
• Example:
  1. The toss of a (fair) coin: S = {H, T}, where H and T denote ‘heads’ and ‘tails’, respectively, and are called the elements or members of the sample space.
  2. The score of a (fair) die: S = {1, 2, 3, 4, 5, 6}.
• So the coin toss sample space has two elementary outcomes, H and T, while the score on a die has six elementary outcomes. These individual elementary outcomes are themselves events, but we may wish to consider slightly more exciting events of interest. For example, for the score on a die, we may be interested in the event of obtaining an even score, or a score greater than 4 etc. Hence we proceed to define an event of interest.
• Typically, we can denote events by letters for notational efficiency. For example, A = ‘an even score’ and B = ‘a score greater than 4’. Hence A = {2, 4, 6} and B = {5, 6}.
• The universal convention is that we define probability to lie on a scale from 0 to 1 inclusive.
• Hence the probability of any event A, say, is denoted P (A) and is a real number somewhere in the unit interval, i.e. P(A) ∈ [0, 1], where ‘∈’ means ‘is a member of’. Note the following.
  • If A is an impossible event, then P(A) = 0.
  • If A is a certain event, then P(A) = 1.
  • For events A and B, if P(A) > P(B), then A is more likely to occur than B.

CAT Preparation: Equally Likely Elementary Outcomes

Classical probability is a simple special case where values of probabilities can be found by just counting outcomes. This requires that:

• the sample space contains only a finite number of outcomes, N
• all of the outcomes are equally probable (equally likely).

Standard illustrations of classical probability are devices used in games of chance: tossing a fair coin (heads or tails) one or more times

• rolling one or more fair dice (each scored 1, 2, 3, 4, 5 or 6)
• drawing one or more playing cards at random from a deck of 52 cards.

Suppose that the sample space S contains N equally likely outcomes, and that event A consists of n ≤ N of these outcomes. We then have that:

• P (A) = n/N = Number of outcomes in A/total number of outcomes in the sample space S.
• That is, the probability of A is the proportion of outcomes which belong to A out of all possible outcomes.
• In the classical case, the probability of any event can be determined by counting the number of outcomes which belong to the event, and the total number of possible outcomes.

Example

• For the coin toss, if A is the event ‘heads’, then N = 2 (H and T) and n = 1 (H). So, for a fair 2 coin, P(A) = 1/2 = 0.5.3
• For the die score, if A is the event ‘an even score’, then N = 6 (1, 2, 3, 4, 5 and 6) and n = 3 (2, 4 and 6). So, for a fair die, P(A) = 3/6 = 1/2 = 0.5. Finally, if B is the event ‘score greater than 4’, then N = 6 (as before) and n = 2 (5 and 6). Hence P(B) = 2/6 = 1/3.

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CAT Preparation: Probability Distribution

A natural question to ask is ‘what is the probability of any of these values?’. That is, we are interested in the probability distribution of the experimental random variable. Be aware that random variables come in two varieties – discrete and continuous. 

• Discrete: Synonymous with ‘count data’, that is, random variables which take non-negative integer values, such as 0, 1, 2, . . .. For example, the number of heads in n coin tosses.
• Continuous: Synonymous with ‘measured data’ such as the real line, R = (−∞, ∞), or some subset of R, for example the unit interval [0, 1]. For example, the height of adults in centimetres.
• The mathematical treatment of probability distributions depends on whether we are dealing with discrete or continuous random variables. We will tend to focus on discrete random variables for much of this course.
• In most cases there will be a higher chance of the random variable taking some sample space values relative to others. Our objective is to express these chances using an associated probability distribution. In the discrete case, we can associate with each ‘point’ in the sample
• space a probability which represents the chance of the random variable being equal to that particular value. (The probability is typically non-zero, although sometimes we need to use a probability of zero to identify impossible events.)
• To summarise, a probability distribution is the complete set of sample space values with their associated probabilities which must sum to 1 for discrete random variables. The probability distribution can be represented diagrammatically by plotting the probabilities against sample space values.

CAT Preparation: Probability Using Venn Diagram

• The sum of all the values in the diagram is 1. 
• The diagram represents the sample space. 
• P(A) – To find the P(A), we will add the probability that only A occurs to the probability that A and B occur to get 0.4+0.3=0.7. So P(A)=0.7.
• P(B) – Similarly, P(B)=0.2+0.3=0.5.
• P(A∩B) – Now, P(A∩B) is the value in the overlapping region 0.3.
• P(A∪B)
  • P(A∪B)=0.4+0.3+0.2=0.9. This can also be found using the formula
  • P(A)+P(B)−P(A∩B)=0.7+0.5−0.3=0.9.

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CAT Venn Diagram for Probability Sample Question & Answer

In the below section, some sample questions along with the answer for CAT Venn Diagram for Probability are given below. 

Ques. Among 60 candidates in a class, someone who wants to select math is also free to choose physics. No candidates want to study both chemistry and physics except only 60 candidates. Every candidate studies at least one subject among the three. However, the number of candidates who study one subject is greater than the number of candidates who studied more than one subject. Find out the maximum and a minimum number of candidates who want to study the subject Chemistry only?

1. 40, 0
2. 28, 0
3. 38, 2
4. 44, 0

Correct Answer: 4

Ques. Birthdate of Harry is Feb 29th of 2012 that falls on Wednesday. If Harry lived longer i.e 101 years, then what are the number of birthdays he will celebrate only on Wednesday? 

1. 3
2. 4
3. 5
4. 1

Correct Answer: 2

Ques. What number of statements from the following section are correct? 

• Any year does not contain 5 Sundays only in the month of May and also total 5 Thursdays in the month of June.

• If Feb 14 of a single year falls in Friday and May 14 also falls in the same year that cannot be a Thursday

• If there are a total of 53 Sundays in a year and it contains 5 Mondays in the month of May. 

1. 0
2. 1
3. 2
4. 3

Correct Answer: 2

Ques. Set P contains all multiples of 4 that are less than 500. Set Q consists of all odd multiples of 7 that is less than 500. On the other side, Set R covers all multiples of 6 less than 500. What are the number of elements that are present in P ∪ Q ∪ R?

1. 202
2. 243
3. 228
4. 186

Correct Answer: 1

Ques. There are 95% candidates in a class who have taken the subject Marketing whereas 80% have selected Finance. On the other side, 84% and 90% have selected for operations (ops), and Human Resources (HR) respectively. Find out the maximum and minimum percentage of candidates who have selected all of the four.

1. 80% and 56%
2. 95% and 53%
3. 80% and 49%
4. 80% and 51%

Correct Answer: 3

Ques. Set A contains all three digit numbers and the numbers are multiples of 5. And Set B consists of all three–digit numbers that are even and multiples of 3. On the other side, Set C covers all three–digit numbers which are multiples of 4. What are the number of elements that are presented in A ∪ B ∪ C?

1. 420
2. 405
3. 555
4. 480

Correct Answer: 1

Ques. If Set A = {2, 3, 5, 6, 7} and Set B = {a, b, c}. What are the number of onto functions are explained from Set B to Set A?

1. 2
2. 3
3. 4
4. None of the above

Correct Answer: 4

Ques. Rocky set up a new business. And for this, he opened two accounts in two different banks (i.e. Axis and SBI). Between two banks, he deposited his earned money for each day in a bank. But, he does not deposit his money in both the banks simultaneously on a given day. But he could not run his business for a long time. Hence, he had to close his business. How many days Rocky carried on the business if…

• He did not deposit in axis bank for 20 days and in SBI for 24 days.

• He deposited in either Axis bank or SBI in 28 days.

1. 36
2. 18
3. 13
4. 24

Correct Answer: 1

Ques.There are a total of 150 students in a class that is from 1 to 150 , among them the candidates from all the even numbered are pursuing CA, the candidate whose number is divided by 5 are pursuing Actuarial. And the candidates whose numbers are divided by 7 are taking preparation for the MBA program. Find out the number of candidates who are doing nothing. 

1. 37
2. 45
3. 51
4. 62

Correct Answer: 3

Ques. If set A and set B are bijective and set C and also set D are bijective, Mark the correct one if there exist a bijection between AC + BD or not 

1. Yes
2. No
3. Data insufficient
4. Cannot be determined

Correct Answer: 1

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