Probability Questions in GATE CSE: Bayes, Expectation, Traps

Probability is the most reliable part of GATE CSE mathematics. Conditional probability and Bayes theorem score 94.0 out of 100 in the book's evidence model, tied with eigenvalues for the top of the section, and a conditional probability item appears in some form in essentially every paper reconstructed from 2010 to 2026. This guide covers the four patterns that generate almost all of it.

Key takeaways

• Four patterns cover the ground: conditional probability and Bayes, expectation and variance, the named distributions, and descriptive statistics.
• Bayes questions change their story every year and keep their arithmetic. Practise the arithmetic.
• The base-rate trap is deliberate. A highly accurate test on a rare condition still produces mostly false positives.
• Almost all of these items are numeric answer type, so an attempt cannot cost you marks.
• Descriptive statistics is named in the syllabus but scores only 53.0. It is cheap to prepare and should not eat Tier 1 time.

What the syllabus actually names

The official GATE 2027 CS syllabus lists, under Probability and Statistics, random variables, the uniform, normal, exponential, Poisson and binomial distributions, mean, median, mode and standard deviation, and conditional probability and Bayes theorem. That wording is unchanged from 2026.

Nothing beyond that list is in scope. Regression, hypothesis testing and confidence intervals are not named and have not been tested in the recent papers. Confirm the current syllabus yourself at gate2027.iitm.ac.in before you plan around it.

Years are an analyst reconstruction from public archives and are approximate. Scores are model based analytical confidence values, not probabilities that a question will appear in 2027.

Pattern 1: conditional probability and Bayes

The template has barely changed in twenty years. Two or three sources contribute known shares of a population, each with its own error or defect rate, an item is drawn and found defective, and you are asked which source it probably came from. The framing rotates through machines, suppliers, students, urns, dice and medical tests.

The method never rotates. Write the posterior as the one path you care about divided by the sum over every path:

A factory takes 60 percent of its parts from supplier A with a 2 percent defect rate, and 40 percent from supplier B with a 5 percent defect rate. A part is found defective. The probability it came from B is 0.4 times 0.05, divided by 0.6 times 0.02 plus 0.4 times 0.05, which is 0.02 divided by 0.032, or 0.625.

Two habits protect you. Write the denominator in full before dividing anything, and label which conditional you have been given. The single most repeated trap in this pattern is answering with the probability of the evidence given the source when the question asked for the source given the evidence.

Pattern 2: the base-rate trap

This is Pattern 1 with numbers chosen so that intuition fails. Suppose a test detects a condition in 99 percent of the people who have it and returns a negative result for 95 percent of those who do not, and the condition affects 1 person in 1,000. A person tests positive. The posterior is 0.001 times 0.99 divided by the sum of 0.001 times 0.99 and 0.999 times 0.05, which is about 0.0194, or under 2 percent.

The reason is the base rate. Out of 100,000 people, about 99 true positives are swamped by about 4,995 false positives. Nothing about the test is faulty; the prior is doing the work. If a GATE probability answer looks surprising, check whether the prior was small before you assume you made an arithmetic error.

The book scores all 25 Engineering Mathematics concepts on the same model and pairs each probability pattern with worked practice questions covering its variants.

Pattern 3: expectation and variance

Linearity of expectation is the highest-value idea in this pattern because it needs no independence. Define an indicator variable for each event you want to count, take the expectation of each, and add them.

A fair die is rolled six times. What is the expected number of faces that never appear? For each face, the probability it is missed on all six rolls is (5/6)^6. Summing six such indicators gives 6 times (5/6)^6, which is about 2.01. The same trick counts expected fixed points of a permutation, expected matches, and expected hash collisions.

For variance, three identities carry almost everything: the variance equals the expectation of the square minus the square of the expectation, scaling by a constant multiplies the variance by the square of that constant, and for independent variables the variances add whether you take the sum or the difference.

A quick check: if X is uniform on the interval from 2 to 8, its variance is 36 divided by 12, which is 3, and its mean is 5, so the expectation of X squared is 3 plus 25, which is 28. Notice that the variance identity, not integration, did the work.

Pattern 4: the named distributions

Each named distribution contributes one or two standard computations.

• Uniform. Mean is the midpoint; variance is the square of the width divided by 12; interval probabilities are ratios of lengths.
• Exponential. Mean is 1 divided by the rate, and the distribution is memoryless. If a component's lifetime is exponential with a mean of 4 hours, the probability it survives 2 more hours given that it has already lasted 3 is just the probability it survives 2 hours from new, which is e raised to minus 0.5, about 0.607.
• Poisson. Mean equals variance equals the rate times the window. Rescaling the rate to the window asked is where marks are lost. If requests arrive at 6 per hour, the rate for a 10-minute window is 1, so the probability of exactly 2 requests is e to the minus 1 divided by 2, about 0.184.
• Binomial. Mean is n times p, variance is n times p times one minus p, and the questions ask for at-least-k probabilities.
• Normal. Rarely more than symmetry, the fact that mean, median and mode coincide, and manipulation of the standard normal.

Geometric waiting times sit alongside these. The expected number of trials to a first success is the reciprocal of the success probability, which is the backbone of the "a die is thrown until" family.

The traps that repeat

These sit alongside every other subject's traps in the common mistakes guide, and the formulas appear in the GATE CSE formula sheet.

How to practise probability for the exam

Work in the format the paper uses. Since most of this is numeric answer type, practise typing three-decimal answers and keep a written record of where your rounding differs from the key, as described in the MCQ, MSQ and NAT strategy guide.

Build a single-page card with the Bayes template, the indicator trick, the variance identity and the five distribution lines, and rewrite it from memory once a week. Then read the rest of the section's priorities in Engineering Mathematics important topics and check where probability sits in the wider most repeated topics ranking.