chapter

Independence and Expected Waiting Time PYQs for GATE DA

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

Try a question

Answer it here to see how it works. Nothing is recorded until you sign in.

Question 1

2024 PYQ

Level 3: Exam Standard

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?

Free preview ends here

Login to view the complete previous-year questions and solutions

Creating an account is free. You get the rest of this chapter, step-by-step solutions, and a study plan built around the topics you are actually weak at.

Why MastersUp

Personalised first. High quality throughout.

Most platforms hand everyone the same content. Here the content moves with your performance, topic by topic.

Built around you, not around a syllabus PDF

Every answer you give moves your topic-level intelligence rate. The next question, the next revision card and tomorrow's plan all change with it.

Revision that hits your weak spots

We only revise topics you have actually attempted and are still below the safe bar on — never the same chapter on repeat.

Questions calibrated to the real exam

Each question carries a measured toughness. You are served a rung above your current level, so practice keeps stretching you.

Notes written for recall, not for volume

Full lesson cards for first study, curated short-note cards for the last mile — with derivations, traps and exam patterns marked.

One place for everything

Notes, chapter practice, previous-year questions, test series and full-length papers — all feeding one picture of your preparation.

Honest progress

No vanity streaks. Progress here means chapters mastered and accuracy that held up on harder questions.

Unlock the whole course

Full notes and short notes, the complete question bank with worked solutions, mock tests, full-length papers, and an adaptive plan that rebuilds itself as you improve.

View all plans Start free

Independence and Expected Waiting Time PYQs for GATE DA

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 · 2024 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

Correct Answer:

Step-by-Step Solution

Insight: This is a "wait for a pattern" question — two consecutive evens. The trials are independent, so track progress toward the pattern using states, and write one equation per state.

Exam route:

Let $p = 1/2$ (probability of even on a fair die).

States: $S_{0}$ (no progress / last was odd), $S_{1}$ (last throw was even), $S_{2}$ (done — two consecutive evens).

Let $E_{i}$ = expected additional throws from state $S_{i}$ .

From $S_{0}$ : throw once. With prob $p$ get even $\to S_{1}$ ; with prob $1 - p$ get odd $\to S_{0}$ .

$E_{0} = 1 + p E_{1} + (1 - p) E_{0}$

From $S_{1}$ : throw once. With prob $p$ get even $\to S_{2}$ (done, 0 more); with prob $1 - p$ get odd $\to S_{0}$ .

$E_{1} = 1 + p \cdot 0 + (1 - p) E_{0}$

Substitute $p = 1/2$ :

$E_{0} = 1 + \frac{1}{2} E_{1} + \frac{1}{2} E_{0} \Rightarrow \frac{1}{2} E_{0} = 1 + \frac{1}{2} E_{1} \Rightarrow E_{0} = 2 + E_{1}$

$E_{1} = 1 + \frac{1}{2} E_{0}$

Substitute $E_{0} = 2 + E_{1}$ into the second:

$E_{1} = 1 + \frac{1}{2} (2 + E_{1}) = 2 + \frac{1}{2} E_{1} \Rightarrow \frac{1}{2} E_{1} = 2 \Rightarrow E_{1} = 4$

$E_{0} = 2 + 4 = 6$ .

Answer: 6 (option C).

Learning route:

A fair die has 3 even faces (2, 4, 6) and 3 odd faces (1, 3, 5), so $p = P (even) = 1/2$ .

We want the expected number of throws until we see two evens in a row.

Define states by how much of the target pattern we have completed:

$S_{0}$ : no useful progress (we are at the start, or the last throw was odd).
$S_{1}$ : the last throw was even (we are one step into the pattern).
$S_{2}$ : we have just seen two consecutive evens (absorbing state, we stop).

Let $E_{i}$ be the expected number of additional throws needed to reach $S_{2}$ starting from $S_{i}$ .

From $S_{0}$ , we spend 1 throw. With probability $1/2$ we get even and move to $S_{1}$ ; with probability $1/2$ we get odd and stay in $S_{0}$ . So $E_{0} = 1 + \frac{1}{2} E_{1} + \frac{1}{2} E_{0}$ .

From $S_{1}$ , we spend 1 throw. With probability $1/2$ we get even and reach $S_{2}$ (done, 0 more throws); with probability $1/2$ we get odd and fall back to $S_{0}$ (all progress lost). So $E_{1} = 1 + \frac{1}{2} (0) + \frac{1}{2} E_{0}$ .

Solving the first equation: $E_{0} - \frac{1}{2} E_{0} = 1 + \frac{1}{2} E_{1} \Rightarrow E_{0} = 2 + E_{1}$ .

Substituting into the second: $E_{1} = 1 + \frac{1}{2} (2 + E_{1}) = 2 + \frac{1}{2} E_{1} \Rightarrow E_{1} = 4$ .

Therefore $E_{0} = 2 + 4 = 6$ .

The expected number of throws is 6.

Wrong paths:

Option A (2): uses $E [T] = 1/ p = 2$ . This is the expected wait for a SINGLE even, not two consecutive.
Option B (4): uses $E [T] = 1/ p^{2} = 4$ . This treats two consecutive evens as two independent events, ignoring the sequential overlap of attempts.
Option D (8): uses $2/ p^{2} = 8$ . This incorrectly doubles the $1/ p^{2}$ formula, perhaps thinking "two evens means multiply by 2".

Try a question

Login to view the complete previous-year questions and solutions

Personalised first. High quality throughout.

Built around you, not around a syllabus PDF

Revision that hits your weak spots

Questions calibrated to the real exam

Notes written for recall, not for volume

One place for everything

Honest progress

Unlock the whole course

Independence and Expected Waiting Time PYQs for GATE DA

Chapter Roadmap: Independence and Expected Waiting Time

The Big Question: How Long Do I Wait?

The setup

Two worlds

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

Step-by-Step Solution

More previous year questions (pyqs) in this unit