---
title: "Poker test, 1000 four-digit numbers (2072 Magh)"
subject: "Simulation and Modeling"
chapter: 6
key: "semester_8/simulation_and_modeling:poker-test-numerical-2072-magh"
source: https://www.bctengineer.com/q/semester_8/simulation_and_modeling/poker-test-numerical-2072-magh
site: BCT Engineering
---

# Poker test, 1000 four-digit numbers (2072 Magh)

Simulation and Modeling · Chapter 6: Random Number

- Asked in: 2072 Magh, 8 marks
- Video: [Poker Test (4 digits) | Random Number Independency | Simulation & Modelling | in Nepali](https://www.youtube.com/watch?v=BNIt43dxQaY) — Monkey D. Luffy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_8/simulation_and_modeling/poker-test-numerical-2072-magh
- Whole chapter: https://www.bctengineer.com/md/semester_8/simulation_and_modeling/chapter-6

## Question

A sequence of 1000 four digit numbers has been generated and an analysis indicates the following conditions. Based on poker test, are these numbers independent? (use α=0.05 and N = 4 is 9.488)

| Combination Distributions | Observed Frequency |
|---|---|
| 4 different Digits | 510 |
| 3 Like Digits | 40 |
| 4 Like Digits | 5 |
| 1 Pair | 410 |
| 2 Pair | 35 |

## Short answer

Using standard four‑digit probabilities, χ0² = 20.01. With 4 degrees of freedom and critical value 9.488 at α=0.05, since 20.01 > 9.488, the numbers are not independent.

## Answer

**Setup.** H0: the numbers are independent. H1: they are not. N = 1000, five patterns, df = 4, χ²0.05,4 = 9.488. The classes are rearranged in the usual order: 4 different, 1 pair, 2 pair, 3 like, 4 like.

Probabilities for four-digit numbers: 0.504, 0.432, 0.027, 0.036, 0.001.

| Pattern | Probability | Oi | Ei | (Oi − Ei)²/Ei |
|---|---|---|---|---|
| 4 different digits | 0.504 | 510 | 504 | 0.071 |
| 1 pair | 0.432 | 410 | 432 | 1.120 |
| 2 pair | 0.027 | 35 | 27 | 2.370 |
| 3 like digits | 0.036 | 40 | 36 | 0.444 |
| 4 like digits | 0.001 | 5 | 1 | 16.000 |

**χ0² = 20.01**

**Conclusion.** Since 20.01 > 9.488, H0 is **rejected**; the numbers are not independent according to the poker test. Most of the value comes from "4 like digits" (5 observed against 1 expected, contribution 16.0).

*Remark:* if the small classes (3 like and 4 like, Ei = 37) were merged, df = 3 and the statistic would be 5.29 < 7.815, so independence would be accepted. With the critical value 9.488 given, the answer is to reject.

## Check yourself

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

1. Before computing, the classes must be matched to their probabilities. Which is correct for '4 like digits'?
   A. 0.504
   B. 0.432
   C. 0.027
   D. 0.001
   Answer: D. Four alike has probability 10/10^4 = 0.001, so Ei = 1 for N = 1000. (Key point: Order of classes)

2. 4 like digits: observed 5, expected 1. Contribution?
   A. 4
   B. 16
   C. 5
   D. 25
   Answer: B. (5 - 1)^2 / 1 = 16. (Key point: Largest contribution)

3. With five classes at alpha = 0.05, the critical value is:
   A. 12.592
   B. 7.815
   C. 9.488
   D. 16.92
   Answer: C. df = 4 and chi-square 0.05,4 = 9.488. (Key point: Critical value)

4. chi-square0 = 20.01 means:
   A. Not independent (reject H0)
   B. Independent (accept H0)
   C. Uniform
   D. Cannot reject
   Answer: A. 20.01 > 9.488, so the numbers are judged not independent. (Key point: Decision)

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