GATE CS - ENGINEERING MATHEMATICS:Probability & Statistics

Mastering probability & statistics concepts and implementation.

Probability & Statistics for GATE CS

Probability usually contributes about 3–4 marks. Expect conditional probability, Bayes’ theorem, a discrete distribution (binomial / Poisson / geometric), and expectation/variance of a simple random variable. Write the sample space before plugging into a formula.

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(\text{at least one success}) = 1 - P(\text{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)

Tip:

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) = \frac{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) = \frac{P(B \mid A)\,P(A)}{P(B)}

Expanded:

P(A_i \mid B) = \frac{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 shows up often enough that it is worth automatic recall.

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) = \binom{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) = \frac{\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) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}

Standard normal:

Z = \frac{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 = \frac{\sum x_i}{n}

5.2 Median

Middle value when data is sorted.

5.3 Mode

Most frequent value.

5.4 Variance

\sigma^2 = \frac{\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 $P(\text{detected})$ .

P(\text{detected}) = P(\text{corrupted}) \cdot P(\text{detected} \mid \text{corrupted}) = 0.1 \times 0.9 = 0.09

Q2. Bayes’ Theorem (Classic GATE Problem)

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

Find $P(S \mid \text{FREE})$ .

P(S \mid F) = \frac{P(F \mid S)P(S)}{P(F \mid S)P(S) + P(F \mid N)P(N)} = \frac{0.8 \cdot 0.3}{0.8 \cdot 0.3 + 0.1 \cdot 0.7} = \frac{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) = \binom{5}{3} (0.5)^3 (0.5)^2 = \frac{10}{32} = 0.3125

Q4. Poisson Distribution

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

P(0) = \frac{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) = \binom{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) = \frac{P(B \mid A)P(A)}{P(B)}

Expected Value

E[X] = \sum x_i p_i

Variance

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

9. Practice problems

You can add these as premium practice items:

A coin is tossed 10 times. Find $P(X = 5)$ for $X \sim \text{Binomial}(10, 0.5)$ .
A Poisson process has $\lambda = 3$ . Find $P(X > 2)$ .
Compute $P(A \mid B)$ when $P(A \cap B) = 0.2$ , $P(B) = 0.5$ .
Find mean and variance of $X \sim \text{Binomial}(20, 0.25)$ .
A transaction is fraud with probability 0.2. Probability it triggers a warning is 0.7 if fraud and 0.1 if not. Find $P(\text{fraud} \mid \text{warning})$ .

Progress

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