Probability & Statistics – GATE CS Premium Module

Probability & Statistics carries 3–4 high-impact marks in GATE CS every year.

Questions are predictable—mostly involving conditional probability, Bayes’ theorem, discrete distributions, and expectation/variance.

With the right shortcuts, this becomes one of the highest-accuracy topics.

1. Basic Probability (Foundation for 80% of Questions)

Probability tells how likely an event is.

(0 le P(A) le 1)
Favourable outcomes → numerator
Total outcomes → denominator

1.1 Complement Rule

[

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

]

Useful when it’s easier to compute “not happening” than “happening”.

GATE Pattern:

Probability of “at least one success”:

[

P( ext{at least one success}) = 1 - P( ext{no success})

]

1.2 Addition Rule

[

P(A cup B) = P(A) + P(B) - P(A cap B)

]

If (A) and (B) are mutually exclusive:

[

P(A cup B) = P(A) + P(B)

]

1.3 Multiplication Rule

For independent events:

[

P(A cap B) = P(A),P(B)

]

GATE Trick:

Two events are independent iff:

[

P(A mid B) = P(A)

]

2. Conditional Probability

Conditional probability measures the chance of an event given that another has already occurred.

[

P(A mid B) = rac{P(A cap B)}{P(B)}

]

Most GATE probability questions revolve around this idea.

2.1 Bayes’ Theorem (Most Tested Formula)

[

P(A mid B) = rac{P(B mid A),P(A)}{P(B)}

]

Expanded:

[

P(A_i mid B) = rac{P(B mid A_i),P(A_i)}{sum_j P(B mid A_j),P(A_j)}

]

Where GATE uses this:

Naive Bayes classifier
Spam filtering
Fraud detection
Probability-based algorithms

This theorem appears as a guaranteed question every 1–2 years.

3. Random Variables (Discrete & Continuous)

A random variable (RV) assigns numbers to outcomes of a probabilistic experiment.

3.1 Discrete Random Variable

Values are countable.

Examples:

Number of heads in tosses
Number of packets arriving
Rolling a die

Defined using PMF (Probability Mass Function).

Expected Value (Mean):

[

E[X] = sum_i x_i,P(X = x_i)

]

Variance:

[

operatorname{Var}(X) = E[X^2] - (E[X])^2

]

3.2 Continuous Random Variable

Values are uncountable (e.g. height, time, latency).

Defined using PDF (Probability Density Function).

Expected Value:

[

E[X] = int_{-infty}^{infty} x,f(x),dx

]

4. Probability Distributions (High-Yield Section)

4.1 Binomial Distribution (Most Important Discrete Distribution)

Models “number of successes in (n) independent Bernoulli trials”.

[

P(X = k) = inom{n}{k} p^k (1-p)^{n-k}

]

Parameters:

(n): number of trials
(p): probability of success

Mean:

[

E[X] = np

]

Variance:

[

operatorname{Var}(X) = np(1-p)

]

GATE Usage: coin tosses, packet drops, success/failure models.

4.2 Poisson Distribution (Very Frequent)

Used for counting rare events in a time/space interval.

[

P(X = k) = rac{lambda^k e^{-lambda}}{k!}

]

Parameter:

(lambda): average rate

Mean = (lambda)

Variance = (lambda)

GATE Usage:

Queue arrivals, number of messages per second, number of failures, etc.

4.3 Normal Distribution (Conceptual Questions)

PDF:

[

f(x) = rac{1}{sigmasqrt{2pi}} e^{-rac{(x-mu)^2}{2sigma^2}}

]

Standard normal:

[

Z = rac{X - mu}{sigma}

]

Properties:

Symmetric
Mean = median = mode = (mu)
68–95–99.7 rule (1σ, 2σ, 3σ coverage)

GATE typically tests:

Symmetry and shape
Mean/variance
Transformation to standard normal

5. Statistics (Mean, Variance, SD)

5.1 Mean

[

mu = rac{sum x_i}{n}

]

5.2 Median

Middle value when data is sorted.

5.3 Mode

Most frequent value.

5.4 Variance

[

sigma^2 = rac{sum (x_i - mu)^2}{n}

]

5.5 Standard Deviation

[

sigma = sqrt{sigma^2}

]

6. GATE PYQ‑Style Solved Problems

Q1. Conditional Probability

A data packet is corrupted with probability 0.1.

Given that a packet is corrupted, the probability it gets detected is 0.9.

Find (P( ext{detected})).

[

P( ext{detected}) = P( ext{corrupted}) cdot P( ext{detected} mid ext{corrupted})

= 0.1 imes 0.9 = 0.09

]

Q2. Bayes’ Theorem (Classic GATE Problem)

(P(S) = 0.3) (message is spam)
(P( ext{FREE} mid S) = 0.8)
(P( ext{FREE} mid N) = 0.1), where (N) = not spam

Find (P(S mid ext{FREE})).

[

P(S mid F) = rac{P(F mid S)P(S)}{P(F mid S)P(S) + P(F mid N)P(N)}

= rac{0.8 cdot 0.3}{0.8 cdot 0.3 + 0.1 cdot 0.7}

= rac{0.24}{0.31} approx 0.774

]

Q3. Binomial Distribution

Let (X) = number of heads in 5 tosses of a fair coin ((p = 0.5)).

[

P(X = 3) = inom{5}{3} (0.5)^3 (0.5)^2 = rac{10}{32} = 0.3125

]

Q4. Poisson Distribution

For (lambda = 4), find (P(X = 0)).

[

P(0) = rac{4^0 e^{-4}}{0!} = e^{-4}

]

Q5. Expected Value

RV (X) takes values 1, 2, 3 with probabilities 0.2, 0.5, 0.3.

[

E[X] = 1(0.2) + 2(0.5) + 3(0.3) = 0.2 + 1.0 + 0.9 = 2.1

]

7. Common Mistakes to Avoid

Thinking independence implies mutual exclusivity (they are different concepts)
Misapplying Bayes’ theorem without a proper denominator (normalisation)
Using binomial when Poisson is more appropriate (rare events / arrival processes)
Forgetting that PDF values can exceed 1 (only area under the curve matters)
Confusing variance and standard deviation

These small mistakes often cost 1 mark in GATE.

8. Fast Revision Sheet (Last-Minute Notes)

Binomial

[

P(X = k) = inom{n}{k} p^k (1-p)^{n-k}

]

Mean = (np)
Var = (np(1-p))

Poisson

Mean = (lambda)
Var = (lambda)
Use for “rare events” / counts in intervals

Bayes’ Theorem

[

P(A mid B) = rac{P(B mid A)P(A)}{P(B)}

]

Expected Value

[

E[X] = sum x_i p_i

]

Variance

[