Independence and Expected Waiting Time Short Notes for GATE DA: Concepts, Formulas, Worked Examples & Practice

Independence and Expected Waiting Time short notes for GATE DA: 1 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

The Waiting-Time Toolkit: One-Page Summary

Core facts

Independent trials: every trial has success probability $p$ , no memory.
Single success: $E [T] = \frac{1}{p}$ .
Two consecutive successes: $E [T] = \frac{1 + p}{p ^{2}}$ .

The state method (for any pattern)

Define states by progress toward the pattern.
For each non-absorbing state $S_{i}$ , write:
$E_{i} = 1 + j \sum P (transition S_{i} \to S_{j}) E_{j} .$
Solve the linear system. The answer is $E_{0}$ .

Key traps

Pattern waiting is not memoryless. Your current state matters.
Self-overlapping patterns (like HH) take longer than non-overlapping ones (like HT) of the same length.
Do not use $1/ p^{2}$ for two consecutive successes. The correct answer is $(1 + p) / p^{2}$ .

Exam checklist

Identify $p$ from the problem.
Classify: single event, two consecutive, or arbitrary pattern?
Apply the matching formula or the state method.
Sanity-check: does the answer reduce correctly when $p = 1$ ?

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Independence and Expected Waiting Time Short Notes for GATE DA: Concepts, Formulas, Worked Examples & Practice

Independence and Expected Waiting Time short notes for GATE DA: 1 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice ques

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