Basic Probability, Events and Counting Previous Year Questions (PYQs) for GATE DA: 5+ Solved Questions with Step-by-Step Solutions

Solve 5+ Basic Probability, Events and Counting previous year questions for GATE DA with answers and detailed solutions. Free sample questions below.

Chapter Roadmap: Probability, Events & Counting

Chapter Journey

1. Counting-Based Probability

The Foundation. Learn to count favorable vs total outcomes using permutations and combinations.

2. Coin Toss & Elementary Operations

Defining the Space. Sample spaces, unions, intersections, and complements.

3. Conditional Probability & Independence

Updating Beliefs. How new information changes probability. Bayes' Theorem.

4. Random Variables & Distributions

Quantifying Outcomes. Discrete/continuous variables, expectation, variance.

Why start here? In competitive exams, many problems look like probability questions but are actually counting puzzles in disguise. If you cannot count the sample space correctly, the probability formula yields incorrect results.

The Classical Definition: Probability as a Ratio

The Classical Approach

P (E) = \frac{n ( E )}{n ( S )}

Favorable Outcomes / Total Outcomes

Finite Sample Space: The total number of outcomes must be countable.
Equally Likely: No outcome is favored over another. (Fair coin vs Loaded coin).

The Real Challenge

Calculating $P (E)$ is rarely about plugging numbers. It is almost always a Counting Problem. We use Permutations and Combinations to find $n (S)$ and $n (E)$ .

Basic Probability, Events and Counting: Solved Questions with Step-by-Step Explanations (5 Problems)

Question 1 · Probability and Statistics MCQ

Three fair coins are tossed independently.

T

is the event that two or more tosses
result in heads.

S

is the event that two or more tosses result in tails.
What is the probability of the event

T \cap S

A.
0
B.
0.5
C.
0.25
D.
1

Question 2 · Probability and Statistics MCQ

The probability of a boy or a girl being born is 1/2. For a family having only
three children, what is the probability of having two girls and one boy?

A.
$\frac{3}{8}$
B.
$\frac{1}{8}$
C.
$\frac{1}{4}$
D.
$\frac{1}{2}$

Question 3 · Probability and Statistics MCQ

Let

L = n \to \infty lim k = 0 \sum n \frac{e ^{- n} n ^{k}}{k !}

Which of the following is the value of

L

A.
0.5
B.
1.0
C.
0
D.
$e^{- 1}$

Question 4 · Probability and Statistics MCQ

Let

M

be a randomly chosen non-empty subset of

S = {1, 2, 3, \dots, 2026}

.
Which of the following is the probability that the product of all the elements of

M

is even?

A.
$2^{1013} (2^{1013} - 1) / 2^{2026}$
B.
$2^{1013} / 2^{2026}$
C.
$2^{1013} (2^{1013} - 1) / (2^{2026} - 1)$
D.
$1/ (2^{2026} - 1)$

Question 5 · Probability and Statistics MCQ

Suppose that a computer program provides a non-negative and integer-valued random solution to the equation

n_{1} + n_{2} + n_{3} + n_{4} = 20

.
Which of the following is the probability that all of

n_{1}, n_{2}, n_{3}, n_{4}

in the provided solution are positive?

A.
$(3 19) / (3 23)$
B.
$(4 20) / (4 24)$
C.
$(3 20) / (3 23)$
D.
$(4 19) / (4 24)$

A question from this chapter

Question 1

Three fair coins are tossed independently. T is the event that two or more tosses result in heads. S is the event that two or more tosses result in tails. What is the probability of the event T \cap S ?

Question 2

The probability of a boy or a girl being born is 1/2. For a family having only three children, what is the probability of having two girls and one boy?

Question 3

Let L = n \to \infty lim k = 0 \sum n \frac{e ^{- n} n ^{k}}{k !} Which of the following is the value of L ?

Question 4

Let M be a randomly chosen non-empty subset of S = {1, 2, 3, \dots, 2026} . Which of the following is the probability that the product of all the elements of M is even?

Question 5

Suppose that a computer program provides a non-negative and integer-valued random solution to the equation n_{1} + n_{2} + n_{3} + n_{4} = 20 . Which of the following is the probability that all of n_{1}, n_{2}, n_{3}, n_{4} in the provided solution are positive?

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