PROBABILITY THEORY   Complete Masterclass for Competitive Examinations

 

7 Chapters

50+ Formulas

80+ MCQs

CAT | GATE | GRE | GMAT

 

 

01	Fundamentals & Axioms
02	Conditional Probability & Bayes' Theorem
03	Random Variables & Distributions
04	Expectation, Variance & Moments
05	Special Distributions
06	Joint Distributions & Limit Theorems
07	Practice MCQs with Detailed Solutions

 

Chapter 1

Fundamentals of Probability

 

1.1  Basic Definitions

• Experiment: Any process whose outcome is uncertain (e.g., tossing a coin).
• Sample Space (S): Set of all possible outcomes of an experiment.
• Event (E): A subset of the sample space.
• Mutually Exclusive Events: A and B cannot both occur; A intersection B = empty set.
• Exhaustive Events: Events whose union equals the entire sample space.
• Equally Likely Events: Events with identical probability of occurrence.

 

1.2  Classical Probability

When all outcomes are equally likely, the probability of event E is:

 

P(E) = (Number of favourable outcomes) / (Total outcomes in S)

 

1.3  Axiomatic Probability (Kolmogorov)

Axiom 1: P(E) >= 0  for every event E.

 

Axiom 2: P(S) = 1  (sample space has probability 1).

 

Axiom 3 (Additivity): If A intersection B = empty set, then P(A union B) = P(A) + P(B).

 

1.4  Key Theorems

Addition Rule

P(A union B) = P(A) + P(B) - P(A intersection B)

 

Complement Rule

P(A') = 1 - P(A)

 

Impossible Event

P(empty set) = 0

 

Inclusion-Exclusion (3 events)

P(AUBUC) = P(A)+P(B)+P(C) - P(AintB) - P(BintC) - P(AintC) + P(AintBintC)

 

Exam Tip

In 'at least one' problems, use the complement: P(at least one) = 1 - P(none). This is almost always faster in exam conditions.

 

Counting Techniques Reference

Operation	Formula	Order Matters?	Repetition?
Permutation	nPr = n! / (n-r)!	Yes	No
Combination	nCr = n! / [r!(n-r)!]	No	No
Permutation (rep)	n^r	Yes	Yes
Combination (rep)	(n+r-1)Cr	No	Yes

 

Chapter 2

Conditional Probability & Independence

 

2.1  Conditional Probability

The probability of A given that B has occurred (P(B) > 0):

 

P(A | B) = P(A intersection B) / P(B)

 

2.2  Multiplication Rule

P(A intersection B) = P(A) * P(B | A) = P(B) * P(A | B)

 

P(A intersection B intersection C) = P(A) * P(B|A) * P(C|A intersection B)

 

2.3  Independent Events

A and B are independent if knowledge of one does not affect the other:

 

P(A intersection B) = P(A) * P(B)

Note: Mutually exclusive events (with P > 0) are NOT independent.

 

2.4  Law of Total Probability

If B1, B2, ..., Bn form a partition of S (mutually exclusive & exhaustive):

 

P(A) = P(A|B1)*P(B1) + P(A|B2)*P(B2) + ... + P(A|Bn)*P(Bn)

 

2.5  Bayes' Theorem

Used to 'reverse' conditional probabilities — most tested in competitive exams:

 

P(Bk | A) = P(A|Bk)*P(Bk) / [P(A|B1)*P(B1) + ... + P(A|Bn)*P(Bn)]

 

Worked Example — Disease Testing

Given	Value
Disease prevalence P(D)	0.01
P(+ve | D) — sensitivity	0.99
P(+ve | D') — false positive	0.05
P(D | +ve) via Bayes' Theorem	approximately 0.167 (only 16.7%!)

 

Exam Tip

Bayes' Theorem problems almost always benefit from a tree diagram. List priors on first branches, likelihoods on second — the answer is one product divided by the total.

 

Chapter 3

Random Variables

 

3.1  Discrete Random Variables

A random variable X taking countable values x1, x2, ... with probabilities p1, p2, ...

 

PMF: P(X = xi) = pi,   pi >= 0,   Sum(pi) = 1

 

CDF: F(x) = P(X <= x) = Sum of P(X = xi) for xi <= x

 

3.2  Continuous Random Variables

X is continuous if it has a probability density function (PDF) f(x):

 

f(x) >= 0 for all x;   Integral of f(x) dx = 1

 

P(a <= X <= b) = Integral from a to b of f(x) dx

 

CDF: F(x) = Integral from -inf to x of f(t) dt;   P(X = a) = 0

 

3.3  Expectation & Variance Reference Table

Quantity	Formula
E[X]  (discrete)	Sum of xi * P(X = xi)
E[X]  (continuous)	Integral of x * f(x) dx
E[aX + b]	a*E[X] + b
E[X + Y]	E[X] + E[Y]  (always)
Var(X)	E[X^2] - (E[X])^2
Var(aX + b)	a^2 * Var(X)
Var(X + Y) if indep.	Var(X) + Var(Y)
SD(X)	sqrt(Var(X))

 

Chapter 4

Special Probability Distributions

 

4.1  Binomial Distribution  B(n, p)

n independent Bernoulli trials each with success probability p; X = number of successes.

 

P(X = k) = C(n,k) * p^k * (1-p)^(n-k),   k = 0, 1, ..., n

 

Property	Value
Mean	np
Variance	np(1-p)
Mode	floor((n+1)p)
MGF	(1-p+p*e^t)^n

 

4.2  Poisson Distribution  P(lambda)

Models rare events over a fixed interval; lambda = average rate.

 

P(X = k) = e^(-lambda) * lambda^k / k!,   k = 0, 1, 2, ...

 

Property	Value
Mean	lambda
Variance	lambda
Approximation	Binomial -> Poisson when n large, p small, lambda = np

 

4.3  Geometric Distribution  Geom(p)

X = number of trials until the first success.

 

P(X = k) = (1-p)^(k-1) * p,   k = 1, 2, 3, ...

 

Property	Value
Mean	1/p
Variance	(1-p)/p^2
Memoryless	P(X > m+n | X > m) = P(X > n)

 

4.4  Uniform Distribution  U(a, b)

Every outcome in [a, b] equally likely.

 

f(x) = 1/(b-a)  for a <= x <= b,  else 0

 

Property	Value
Mean	(a+b)/2
Variance	(b-a)^2 / 12
CDF	(x-a)/(b-a)  for a <= x <= b

 

4.5  Normal Distribution  N(mu, sigma^2)

The bell-curve — the most important distribution in statistics.

 

f(x) = (1/(sigma*sqrt(2*pi))) * exp(-(x-mu)^2 / (2*sigma^2))

 

Property	Value
Mean	mu
Variance	sigma^2
Standardisation	Z = (X - mu) / sigma  =>  Z ~ N(0, 1)
Empirical rule	68% within +/-1sigma, 95% within +/-2sigma, 99.7% within +/-3sigma

 

4.6  Exponential Distribution  Exp(lambda)

Models time between events in a Poisson process.

 

f(x) = lambda * e^(-lambda*x)  for x >= 0

 

Property	Value
Mean	1/lambda
Variance	1/lambda^2
Memoryless	P(X > s+t | X > s) = P(X > t)

 

Exam Tip

Always standardise before using the Z-table: Z = (X - mu)/sigma. Normal is symmetric: Phi(-z) = 1 - Phi(z), and P(a < Z < b) = Phi(b) - Phi(a).

 

Chapter 5

Joint Distributions & Limit Theorems

 

5.1  Joint & Marginal Distributions

Concept	Formula
Joint PMF	p(x,y) = P(X=x, Y=y)
Marginal of X	pX(x) = Sum over y of p(x,y)
Marginal of Y	pY(y) = Sum over x of p(x,y)
Independence	p(x,y) = pX(x) * pY(y)  for all (x,y)
Conditional	p(x|y) = p(x,y) / pY(y)
Covariance	Cov(X,Y) = E[XY] - E[X]*E[Y]
Correlation	rho = Cov(X,Y)/(sigma_X * sigma_Y);  -1 <= rho <= 1

 

5.2  Law of Large Numbers (LLN)

As n -> infinity, the sample mean converges in probability to the true mean mu:

 

X_bar_n = (X1 + X2 + ... + Xn) / n  -->  mu  as n --> infinity

 

5.3  Central Limit Theorem (CLT)

Regardless of the original distribution, the sample mean is approximately Normal for large n:

 

sqrt(n) * (X_bar - mu) / sigma  -->  N(0,1)  as n --> infinity

Rule of thumb: CLT approximation is reliable for n >= 30.

 

5.4  Chebyshev's Inequality

A non-parametric bound applicable to any distribution with finite mean and variance:

 

P(|X - mu| >= k*sigma) <= 1/k^2   for any k > 0

 

Exam Tip

Chebyshev gives a guaranteed (conservative) bound. For k=2: at least 75% of data lies within 2 standard deviations. For k=3: at least 88.9%.

 

Chapter 6

Practice MCQs — All Levels

 

Foundational (Q1-10)

Intermediate (Q11-20)

Advanced (Q21-30)

 

Q1. A fair coin is tossed twice. P(at least one Head) = ?

(A) 1/4

(B) 1/2

(C) 3/4

(D) 1

Answer: (C) 3/4   P = 1 - P(TT) = 1 - 1/4 = 3/4

Q2. Two dice are rolled. P(sum = 7) = ?

(A) 1/6

(B) 7/36

(C) 1/12

(D) 1/9

Answer: (A) 1/6   Favourable: {(1,6),(2,5),(3,4),(4,3),(5,2),(6,1)} = 6; P = 6/36 = 1/6

Q3. Bag: 5 red, 3 blue balls. 2 drawn without replacement. P(both red) = ?

(A) 25/64

(B) 5/14

(C) 5/28

(D) 10/56

Answer: (B) 5/14   P = (5/8)*(4/7) = 20/56 = 5/14

Q4. P(A)=0.6, P(B)=0.4, A and B independent. P(A intersection B) = ?

(A) 0.24

(B) 0.10

(C) 0.76

(D) 0.20

Answer: (A) 0.24   P(A intersect B) = P(A)*P(B) = 0.6*0.4 = 0.24

Q5. P(A|B)=0.3, P(B)=0.5. Find P(A intersection B).

(A) 0.6

(B) 0.15

(C) 0.8

(D) 0.2

Answer: (B) 0.15   P(A intersect B) = P(A|B)*P(B) = 0.3*0.5 = 0.15

 

Q6. X ~ B(10, 0.4). E[X] = ?

(A) 4

(B) 2

(C) 6

(D) 2.4

Answer: (A) 4   E[X] = np = 10*0.4 = 4

Q7. X ~ B(10, 0.4). Var(X) = ?

(A) 4

(B) 2.4

(C) 1.6

(D) 6

Answer: (B) 2.4   Var = np(1-p) = 10*0.4*0.6 = 2.4

Q8. X ~ Poisson(3). P(X = 0) = ?

(A) 1/e^3

(B) 3/e

(C) e^3

(D) 0

Answer: (A) 1/e^3   P(X=0) = e^(-3)*3^0/0! = e^(-3) = 1/e^3

Q9. X ~ N(10, 4). P(X < 10) = ?

(A) 0.25

(B) 0.75

(C) 0.50

(D) 1

Answer: (C) 0.50   Normal is symmetric about mean; P(X < mu) = 0.50

Q10. X is uniform on [2, 8]. E[X] = ?

(A) 4

(B) 5

(C) 6

(D) 3

Answer: (B) 5   E[X] = (a+b)/2 = (2+8)/2 = 5

 

 

Q11. Box: 4 defective, 6 good. 3 chosen. P(exactly 2 defective) = ?

(A) 3/10

(B) 6/25

(C) 12/35

(D) 1/5

Answer: (A) 3/10   C(4,2)*C(6,1)/C(10,3) = 6*6/120 = 36/120 = 3/10

Q12. P(A)=0.5, P(B|A)=0.6, P(B|A')=0.3. P(A|B) = ?

(A) 2/3

(B) 0.45

(C) 1/2

(D) 0.6

Answer: (A) 2/3   P(B)=0.5*0.6+0.5*0.3=0.45; P(A|B)=0.30/0.45=2/3

Q13. X ~ Geometric(p=0.25). E[X] = ?

(A) 4

(B) 3

(C) 0.25

(D) 12

Answer: (A) 4   E[X] = 1/p = 1/0.25 = 4

Q14. X ~ Poisson(5). P(X <= 1) = ?

(A) 6e^(-5)

(B) 5e^(-5)

(C) e^(-5)

(D) 1-6e^(-5)

Answer: (A) 6e^(-5)   P(0)+P(1) = e^(-5) + 5e^(-5) = 6e^(-5)

Q15. E[X]=3, E[Y]=4, Cov(X,Y)=2. E[XY] = ?

(A) 14

(B) 12

(C) 10

(D) 6

Answer: (A) 14   Cov(X,Y)=E[XY]-E[X]*E[Y] => E[XY]=2+12=14

 

Q16. X ~ N(50, 25). P(40 < X < 60) = ?

(A) 0.9544

(B) 0.6827

(C) 0.9974

(D) 0.50

Answer: (A) 0.9544   40 and 60 are +/-2sigma from mu=50 (sigma=5); by empirical rule ~95.44%

Q17. P(A union B)=0.8, P(A)=0.5, P(B)=0.6. P(A intersection B) = ?

(A) 0.3

(B) 0.1

(C) 0.4

(D) 0.2

Answer: (A) 0.3   P(A intersect B) = 0.5+0.6-0.8 = 0.3

Q18. Var(X)=9, Var(Y)=16, independent. Var(2X-3Y) = ?

(A) 180

(B) 108

(C) 144

(D) 36

Answer: (A) 180   4*Var(X)+9*Var(Y) = 4*9+9*16 = 36+144 = 180

Q19. X ~ B(6, 0.5). P(X >= 4) = ?

(A) 11/32

(B) 21/64

(C) 15/64

(D) 5/16

Answer: (A) 11/32   [C(6,4)+C(6,5)+C(6,6)]/2^6 = (15+6+1)/64 = 22/64 = 11/32

Q20. CLT: X_bar from n=100, mu=50, sigma=10. P(X_bar > 51) = ?

(A) 0.1587

(B) 0.8413

(C) 0.3413

(D) 0.6587

Answer: (A) 0.1587   SE=10/sqrt(100)=1; Z=(51-50)/1=1; P(Z>1)=1-Phi(1)~0.1587

 

 

Q21. 3 machines produce 50%, 30%, 20% of output. Defect rates: 2%, 3%, 4%. P(defective item from machine 3) = ?

(A) 8/27

(B) 20/45

(C) 4/13

(D) 1/5

Answer: (A) 8/27   P(D)=.5*.02+.3*.03+.2*.04=.027; P(M3|D)=0.008/0.027=8/27

Q22. X,Y ~ independent N(0,1). P(X^2+Y^2 <= 4) = ?

(A) 1-e^(-2)

(B) e^(-2)

(C) 1-e^(-4)

(D) 0.5

Answer: (A) 1-e^(-2)   X^2+Y^2 ~ Chi-sq(2)=Exp(0.5); P(<=4)=1-e^(-2) ~ 0.865

Q23. E[X^2]=13, Var(X)=4. E[X] = ?

(A) 3

(B) +/-3

(C) 9

(D) Cannot determine

Answer: (B) +/-3   Var=E[X^2]-(E[X])^2 => 4=13-(E[X])^2 => (E[X])^2=9 => E[X]=+/-3

Q24. X ~ Poisson(lambda). P(X=1)=P(X=2). Find lambda.

(A) 1

(B) 2

(C) 0.5

(D) 3

Answer: (B) 2   lambda*e^(-lambda) = lambda^2*e^(-lambda)/2 => 1=lambda/2 => lambda=2

Q25. Cov(X,Y)=6, sigma_X=3, sigma_Y=4. Corr(X,Y) = ?

(A) 0.5

(B) 0.75

(C) 2

(D) 0.6

Answer: (A) 0.5   rho = 6/(3*4) = 6/12 = 0.5

 

Q26. X ~ U[0,1]. Y = -ln(X). What distribution is Y?

(A) Uniform

(B) Exponential(1)

(C) Normal

(D) Geometric

Answer: (B) Exponential(1)   CDF: P(Y<=y)=P(-lnX<=y)=P(X>=e^(-y))=1-e^(-y); this is Exp(1)

Q27. n fair coins. P(all heads) <= 0.01. Minimum n = ?

(A) 6

(B) 7

(C) 5

(D) 8

Answer: (B) 7   (1/2)^n <= 0.01 => n >= log2(100) ~ 6.64 => n=7

Q28. P(A)=0.4, P(B)=0.5, P(A' intersect B')=0.2. Are A,B independent?

(A) Yes

(B) No

(C) Insufficient data

(D) Mutually exclusive

Answer: (B) No   P(A intersect B)=0.4+0.5-0.8=0.1; P(A)*P(B)=0.20 != 0.1; not independent

Q29. E[X]=2, E[X^2]=8. Var(3X+5) = ?

(A) 36

(B) 9

(C) 45

(D) 12

Answer: (A) 36   Var(X)=8-4=4; Var(3X+5)=9*Var(X)=9*4=36

Q30. Chebyshev: E[X]=10, Var(X)=4. P(|X-10|>=4) <= ?

(A) 1/4

(B) 1/2

(C) 1/3

(D) 1/16

Answer: (A) 1/4   k=4/sigma=4/2=2; P(|X-mu|>=2sigma) <= 1/k^2 = 1/4

 

 

Chapter 7

Quick Formula Reference Sheet

 

Core Probability Rules

P(A') = 1 - P(A)

 

P(A union B) = P(A) + P(B) - P(A intersection B)

 

P(A|B) = P(A intersection B) / P(B)

 

P(A intersection B) = P(A)*P(B)  [if independent]

 

Bayes: P(Bi|A) = P(A|Bi)*P(Bi) / Sum[P(A|Bj)*P(Bj)]

 

 

Expectation & Variance

E[aX+b] = a*E[X] + b

 

Var(aX+b) = a^2*Var(X)

 

Var(X) = E[X^2] - (E[X])^2

 

Cov(X,Y) = E[XY] - E[X]*E[Y]

 

Correlation rho = Cov(X,Y) / (sigma_X * sigma_Y)

 

 

Key Distributions (Mean | Variance)

Binomial B(n,p):      mu = np          |  sigma^2 = np(1-p)

 

Poisson P(lambda):    mu = lambda      |  sigma^2 = lambda

 

Geometric Geom(p):    mu = 1/p         |  sigma^2 = (1-p)/p^2

 

Uniform U[a,b]:       mu = (a+b)/2     |  sigma^2 = (b-a)^2/12

 

Normal N(mu,sig^2):   Standardise Z=(X-mu)/sigma   Z~N(0,1)

 

Exponential Exp(lam): mu = 1/lambda    |  sigma^2 = 1/lambda^2

 

 

Limit Theorems

LLN:       X_bar_n --> mu  as n --> infinity

 

CLT:       sqrt(n)*(X_bar - mu)/sigma --> N(0,1)  as n --> infinity

 

Chebyshev: P(|X-mu| >= k*sigma) <= 1/k^2

 

Markov:    P(X >= a) <= E[X]/a  (for X>=0, a>0)

 

 

Final Revision Note

Before every exam, review Bayes' Theorem, Binomial and Normal distributions, and the CLT. These three areas account for approximately 50% of probability questions in CAT, GATE, and GRE.