Independence and Expected Waiting Time Practice Questions for GATE DA: 0+ Solved Questions with Step-by-Step Solutions

Solve 0+ Independence and Expected Waiting Time practice 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.

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