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Lesson Posted on 14/10/2019 Exam Coaching/MBA Entrance Coaching

Shantanav Bhowmik

My recent academic accomplishments include admission offers (unreserved category) from MDI Gurgaon and...

SB Page 1PROBABILITY- STUDY NOTESPROBABILITY- the method of finding the likelihood of an event.1. For example let us throw a dice,an unbiased dice that is all faces have equal chances of turning up -1,2,3,4,5,6 on six faces. What is the probability that the number up will be an even number?sample space=1,2,3,4,5,6=>... read more

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PROBABILITY- STUDY NOTES

PROBABILITY- the method of finding the likelihood of an event.

1. For example let us throw a dice,an unbiased dice that is all faces have equal chances of turning up -

1,2,3,4,5,6 on six faces. What is the probability that the number up will be an even number?

sample space=1,2,3,4,5,6=> n(s)=6

even number or event space= 2,4,6=> n(e)=3

n()=number of elements

therefore required probability= n(e)/n(s)=3/6=1/2.

P(E)=N(E)/N(S)

2. Now unbiased 2 dies are thrown simultaneously. What is the probability that sum of their faces is 8?

we know that the minimum value on each die is 1 and maximum value is 6.

we also know that we can get a sum of 8 with integers from 1 to 6 as=3+5/4+4

so the number of ways in which the required sum of 8 can face up is=

Dice1 Dice2

4 4

5 3

3 5

so the required number of cases for or event= 4/4,3/5,5/3= 3 cases.

total number of cases (from method of counting)=6*6=36

P(E)=3/36=1/12

2nd method- what is the probability that 4 will turn up in die 1? It is 1/6 as n(e)=1 & n(s)=6. Similarly for

every other number on an unbiased die.

A sum of 8 will be obtained when we get –{(4 on d1) &( 4 on d2)} OR{ (3 on d1) &( 5 on d2)} OR {(5 on d1)

& (3 on d2)} as listed above.

Therefore the required probability is obtained when we replace & by X (multiplication) and OR by +

(addition). And we write the probability for every individual event in the bracket terms.

so P(e)= { 1/6*1/6}+{1/6*1/6}+{1/6*1/6)=1/12.

3. Consider again an unbiased die .

n(even)=3 since cases{2,4,6)

n(odd)=3 since cases (1,3,5)

As set of cases of odd number and set of cases of even number has no element common even if their number

of elements is same –these two sets are said to be mutually exclusive.

Hence we say that the two events of getting odd numbers and even numbers when an unbiased dice is

thrown is mutually exclusive to each other.

4. N(AUB)=N(A)+N(B)-N(A˄B) from SET theory : S = sample space ,in this case represents all possible

outcomes- the whole box.

here if we are asked to find the probability that either A or B occurs ( it means atleast one of A or B) , we will

take help of the above concept of vein diagram as given above.

P(e)= n(A∪B)/n(s)

A A˄B B

S

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5. Consider the case of an unbiased die again.

Let A be the case of getting a prime ∴ N(A)=(2,3,5)=3 ∴P(A)=N(A)/N(S)=3/6=1/2.

Let B be the case of getting an even no ∴ N(B)=(2,4,6)=3 ∴P(B)=N(B)/N(S)=3/6=1/2.

What if we are asked to find out the probability of event A in case it is given that event B has already

occurred?

In this case we find the the outcomes of B are the that is n(b)=3. In these outcomes we are to find the prime

numbers that are in set A to get our event space. So our event space n(e)=n(A˄B)={2}=1 and our sample

space n(b)=3.

So the req P(e)= N(A˄B)/N(B)= 1/3.

This is written as P(A/B)- probability that A will occur given B has already occurred.

Now as req event space is A˄B; We write P(A/B)=P(A˄B)/P(B). This known as conditional probability.

This can be derived as follows- We know P (A/B)=N(A˄B)/N(B)= *N(A˄B)N(S)+/*N(B)/N(S)+=

P(A/B)=P(A˄B)/P(B).

6. Consider set S =*1,2,3,4………47,48,…..50+

A= Event of getting a multiple of 5 from S= { 5,10,15,……,50+

B= Event of getting a multiple of 2 from S =*2,4,6,8………..50+

now P(A)=10/50 , P(A/B)=5/25

As we see that numerically P(A)=P(A/B) that is the probability of event A has not changed given the

condition that B has already occurred , we say that A & B are independent events. As we know that

P(A/B)=P(A˄B)/P(B). Putting the condition for independent events that is P(A)=P(A/B) we get

P(A).P(B)=P(A˄B)

As A & B are independent events as proven we can also show P(A/ )=P(A).

N:B- If P(A˄B)=P(A).P(B) numerically, A &B are pairwise independent.

Similarly if P(A˄B˄C)=P(A).P(B).P(C) , A,B & C are mutually independent.

7. BAYES THEOREM of Conditional Probability

If we are given the probabilities that B occurs given A1, A2…..An occurs and we are asked to find the

probability that Ai will occur given B has already occurred, when we are given or can calculate P(Ai), we use

Bayes theorem.

Note that A1 A2 A3 …..An are mutually exclusive and collectively exhaustive that is individual event spaces

don’t have any common and all As make up A event space.

Given, Mathematically if P(B/Ai)=Qi and P(Ai)=Pi then

P(Ai/B)= Pi.Qi/𝞢Pi.Qi

Example-

Box 1 contains 3 white and 2 black balls. Box 2 contains 1 white and 4 black balls.A ball is picked from one

of the two boxes, it turns out to be black. Find the probability that it was drawn from box 1.

We are to find probability of selecting box 1 given we have picked a black ball=P(B1/BLACK)

By BAYES THEOREM,

P(B1/BLACK)={P(B1).P(BLACK/B1)}/[{ P(B1).(BLACK/B1)}+ {P(B2).P(BLACK/B2)}]

Hence,

P(BLACK/B1)=Probability of drawing a black ball from box 1= 2/5

P(BLACK/B2)=Probability of drawing a black ball from box 2= 4/5

P(B1)=1/2

P(B2)=1/2

∴ P(B1/BLACK)= (1/2)(2/5)/,*(1/2)(2/5)++*(1/2)(4/5)+-=1/3

™Solution of Bayes theorem problem without Bayes theorem

Consider the given problem above.

Note that since we are given that a black ball has been picked already that reduces out sample space

probability to selecting bag 1 and black ball or selecting bag 2 and black ball while the former is our event

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space probability. So basically it reduces to the probability of the way or ways we can get the desired

outcome or outcomes divided by the summation of all the probabilities of available outcomes , given the pre

condition.

Probability of selecting box 1 given we have picked a black ball=

P(B1/BLACK)= event space probability P(e)/ sample space probability P(s) = event of selecting black ball

from B1/ events of selecting black balls from B1 & B2=event of selecting B1 and black ball from it/[ event of

selecting B1 and black ball from it OR event of selecting B2 and black ball from it]= 1/3

8. Expected Value= E1.M1+E2.M2+…….+En.Mn

Example-

A person tosses a coin. It comes up with head he gets 10 rupees. It comes up with tail he has to pay 5 rupees.

What is his expectation ?

E=P(heads)M(heads)+P(tails)M(tails)= (1/2)10+(1/2)(-5)=2.5

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Comments Lesson Posted on 08/10/2019 Exam Coaching/MBA Entrance Coaching/CAT Coaching

Rahul Ranjan

I have worked in various MBA/BANKING preparation institutes across India. Expert in Concept building...

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Comments Answered on 07/09/2019 Exam Coaching/MBA Entrance Coaching/CAT Coaching

Career Desk

Dear Sambit, Yes, MAT exam can be easily cracked. However, for CAT, it is difficult but not impossible. Plan and strategize and implement. If you are focussing full time for the next three months and practice smartly. You will definetely achieve good percentile. Wishing you all the best.

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Answers 6 Comments Looking for MBA Entrance Coaching classes

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Lesson Posted on 25/07/2019 Exam Coaching/MBA Entrance Coaching/CAT Coaching

Finding highest power of a number in a factorial

Anubhav

Suppose we want to find highest power of a number(n) in n factorial m It is calculated by =++.... where is greatest integer Example if we want to find highest power of 5 in 30 factorial It will be + =6+1 So highest power of 5 in 30 factorial is 7 read more

Suppose we want to find highest power of a number(n) in n factorial m

It is calculated by

=[m/n]+[m/n*n]+[m/n*n*n]....

where [] is greatest integer

Example if we want to find highest power of 5 in 30 factorial

It will be [30/5]+[30/25]

=6+1

So highest power of 5 in 30 factorial is 7

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Comments Lesson Posted on 24/07/2019 Exam Coaching/MBA Entrance Coaching

Anubhav

If a number a + b + c is divided by number n then reminder will be sum of reminder if each one is divided by n Example 3 + 4 + 9 divide by 12 what will be reminder 3+4-3=4 will be the reminder here we have used the concept of negative reminder read more

If a number a + b + c is divided by number n then reminder will be sum of reminder if each one is divided by n

Example

3 + 4 + 9 divide by 12 what will be reminder

3+4-3=4 will be the reminder here we have used the concept of negative reminder

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Comments Answered on 26/05/2019 Exam Coaching/MBA Entrance Coaching/CAT Coaching

Atul Mittal

Most Qualified MBA Home Tutor Teaching in TIME CP from 13 Yrs

Forget about profile. Just try getting as high a percentile as possible in CAT. Written +GD/PI/WAT form the major chunk of the selection process in any top b school.

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Answers 5 Comments Answered on 18/03/2019 Exam Coaching/MBA Entrance Coaching/CAT Coaching

When should I Start my CAT preparations?

Ajay Dhar

Guarantee of better results than most, or FREE Tutoring for Entrance Competitions, IIT, NEET, CAT

if you start one year ago, you need less efforts. one hour a day is sufficient. if you start late, 10 hours a day may be needed. I give one hour coaching for free.

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Answers 1 Comments Answered on 19/07/2019 Exam Coaching/MBA Entrance Coaching

Srinjini A.

Hi Jennifer, please clear your basics first. Get in touch with someone who offers courses on the basics and clears your concepts. Once that is made, your path is clear. I would suggest you start with a lower level of difficulty. I have used Arun Sharma for my preparations and its time consuming but... read more

Hi Jennifer, please clear your basics first. Get in touch with someone who offers courses on the basics and clears your concepts. Once that is made, your path is clear. I would suggest you start with a lower level of difficulty.

I have used Arun Sharma for my preparations and its time consuming but useful. I hold an MBA myself, and I can say you will require patience in your preps. All the Best :) I am open to all kinds of necessary suggestions.

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Answers 13 Comments 1 Answered on 08/06/2018 Exam Coaching/MBA Entrance Coaching Tuition/BBA Tuition

Atul Mittal

Most Qualified MBA Home Tutor Teaching in TIME CP from 13 Yrs

BBA should be better in making you understand about a business. But engineering from a top college will help you fine tune your analytical skills.

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Answers 88 Comments Looking for MBA Entrance Coaching classes

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Answered on 30/01/2016 Exam Coaching/MBA Entrance Coaching/CAT Coaching

D.C.G Academy

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