---
title: "Expected tosses until a six"
subject: "Probability and Statistics"
chapter: 2
key: "semester_5/probability_and_statistics:fair-die-expected-tosses-six"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/fair-die-expected-tosses-six
site: BCT Engineering
---

# Expected tosses until a six

Probability and Statistics · Chapter 2: Discrete Probability Distribution

- Asked in: 2075 Chaitra, 5 marks
- Video: [How many times must you roll a die  to get a six?](https://www.youtube.com/watch?v=xH89DaEI35w) — Visually Explained
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/fair-die-expected-tosses-six
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-2

## Question

A fair dice was rolled until one gets a Six; find the expected number of toss required?

## Short answer

The number of tosses until a six follows a geometric distribution with p=1/6, so expected value = 1/p = 6 tosses.

## Answer

Let X be the number of tosses needed to get the first six. Each toss is an independent trial with success probability p = 1/6, so X follows a **geometric distribution**:

P(X = x) = q^(x−1) p = (5/6)^(x−1) (1/6), x = 1, 2, 3, …

#### Expected value
E(X) = Σ x · q^(x−1) p = p (1 + 2q + 3q² + …) = p · 1/(1 − q)² = p / p² = 1/p

E(X) = 1 / (1/6) = **6**

Variance (for reference) = q/p² = (5/6)/(1/36) = 30.

**Answer: on average 6 tosses are required to get a six.**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Which distribution models tosses until the first six?
   A. Binomial
   B. Geometric
   C. Poisson
   D. Hypergeometric
   Answer: B. Independent trials, p = 1/6, count until first success. (Key point: Distribution)

2. What is E(X) for geometric?
   A. 1/p
   B. q/p
   C. np
   D. p
   Answer: A. E(X) = 1/p = 6. (Key point: Expected value)

3. Expected tosses to get a six?
   A. 5
   B. 6
   C. 1/6
   D. 36
   Answer: B. 1/(1/6) = 6. (Key point: Answer)

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