Independence and Expected Waiting Time Previous Year Questions (PYQs) for GATE DA: 1+ Solved Questions with Step-by-Step Solutions

Solve 1+ Independence and Expected Waiting Time previous year questions for GATE DA with answers and detailed solutions. Free sample questions below.

Chapter Roadmap: Independence and Expected Waiting Time

Your journey through this chapter

Independent Trials — the foundation

What "no memory" really means, and why it matters.

Waiting for a single event

The geometric distribution and the clean formula

E [T] = 1/ p

Why patterns break the simple formula

Two consecutive successes is not the same as two independent successes.

The state method — your main weapon

Define states by progress, write equations, solve.

Exam-ready patterns and traps

Recognising the question type, avoiding the memoryless trap, and the HH vs HT surprise.

End goal: given any "repeat until you see pattern X" question, you will set up the states and solve for the expected time in under two minutes.

The Big Question: How Long Do I Wait?

The setup

An experiment is repeated, independently, forever.
Each trial has a success probability $p$ (constant across trials).
You are waiting for some target: a single success, two successes in a row, a specific sequence, etc.

The question
Let

T

be the number of trials until the target is first achieved. What is

E [T]

Two worlds

Waiting for...	Tool	Typical answer shape
A single success	Geometric distribution	$E [T] = \frac{1}{p}$
A pattern (e.g. two in a row)	State equations	$E [T] = \frac{1 + p}{p ^{2}}$

The first world is one line. The second world is where the real exam questions live.

Independence and Expected Waiting Time: Solved Questions with Step-by-Step Explanations (1 Problems)

Question 1 · Probability and Statistics MCQ

A fair six-sided die (with faces numbered 1, 2, 3, 4, 5, 6) is repeatedly thrown
independently.
What is the expected number of times the die is thrown until two consecutive throws
of even numbers are seen?

A.
2
B.
4
C.
6
D.
8

A question from this chapter

Question 1

A fair six-sided die (with faces numbered 1, 2, 3, 4, 5, 6) is repeatedly thrown independently. What is the expected number of times the die is thrown until two consecutive throws of even numbers are seen?

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