---
title: "Poker test, 10,000 five-digit numbers (2072)"
subject: "Simulation and Modeling"
chapter: 6
key: "semester_8/simulation_and_modeling:poker-test-numerical-2072-ashwin"
source: https://www.bctengineer.com/q/semester_8/simulation_and_modeling/poker-test-numerical-2072-ashwin
site: BCT Engineering
---

# Poker test, 10,000 five-digit numbers (2072)

Simulation and Modeling · Chapter 6: Random Number

- Asked in: 2072 Ashwin, 8 marks
- Video: [Poker test for 5 digits in Nepali | Simulation and modeling](https://www.youtube.com/watch?v=6-jvBeNAV6w) — Shrijan Sainju
- 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-ashwin
- Whole chapter: https://www.bctengineer.com/md/semester_8/simulation_and_modeling/chapter-6

## Question

A sequence of 10,000 five digit numbers has been generated and analysis indicates the following combinations and frequencies.

| Combinations (i) | Observed Frequency (Oi) |
|---|---|
| All different | 3450 |
| One pair | 4510 |
| Two pair | 1150 |
| Three of a kind | 750 |
| Full house | 85 |
| Four of a kind | 40 |
| Five of a kind | 15 |
| Total | 10,000 |

Based on Poker Test check whether the numbers are independent. Use α = 0.05 and N = 6 is 12.592

## Short answer

For five‑digit numbers, expected frequencies from probabilities give χ0² = 318.37. With 6 degrees of freedom and α=0.05 critical value 12.592, since 318.37 > 12.592, the null hypothesis of independence is rejected.

## Answer

**Setup.** H0: the numbers are independent. H1: they are not. N = 10,000, 7 classes, so df = 6 and χ²0.05,6 = 12.592.

For five-digit numbers, counting the 10⁵ equally likely strings gives the probabilities: all different 0.3024, one pair 0.5040, two pairs 0.1080, three of a kind 0.0720, full house 0.0090, four of a kind 0.0045, five of a kind 0.0001 (they sum to 1).

**Expected frequency** Ei = 10,000 × probability.

| Pattern | Probability | Oi | Ei | (Oi − Ei)²/Ei |
|---|---|---|---|---|
| All different | 0.3024 | 3450 | 3024 | 60.012 |
| One pair | 0.504 | 4510 | 5040 | 55.734 |
| Two pair | 0.108 | 1150 | 1080 | 4.537 |
| Three of a kind | 0.072 | 750 | 720 | 1.250 |
| Full house | 0.009 | 85 | 90 | 0.278 |
| Four of a kind | 0.0045 | 40 | 45 | 0.556 |
| Five of a kind | 0.0001 | 15 | 1 | 196.000 |

**χ0² = 318.37**

**Conclusion.** Since 318.37 > 12.592, H0 is **rejected**. The numbers are not independent. The large deviations are in "all different" (3450 against 3024 expected), "one pair" (4510 against 5040), and "five of a kind" (15 against 1). Even after merging the two smallest classes, the statistic stays far above the critical value.

## Check yourself

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

1. For N = 10,000 five-digit numbers, what is the expected count of one pair (probability 0.504)?
   A. 504
   B. 5040
   C. 4510
   D. 3024
   Answer: B. Ei = 10,000 x 0.504 = 5040. (Key point: Expected frequencies)

2. Five of a kind: observed 15, expected 1. What is the contribution?
   A. 14
   B. 196
   C. 15
   D. 225
   Answer: B. (15 - 1)^2 / 1 = 196. (Key point: Five of a kind contribution)

3. All different: observed 3450, expected 3024. What is the contribution?
   A. About 60.0
   B. About 426
   C. About 0.14
   D. About 5.6
   Answer: A. 426^2 / 3024 = 181476/3024 = 60.0. (Key point: All different contribution)

4. chi-square0 = 318.37 vs 12.592 means:
   A. Accept independence
   B. Reject independence
   C. Merge classes and accept
   D. Test inconclusive
   Answer: B. 318.37 > 12.592, so H0 is rejected; merging small classes does not change this. (Key point: Decision)

## Close variants set in other exams

- [Poker test, 10,000 five-digit numbers (2074)](https://www.bctengineer.com/question.md?key=semester_8%2Fsimulation_and_modeling%3Apoker-test-numerical-2074-magh) — Simulation and Modeling, chapter 6, asked 2×
- [Poker test, 1000 four-digit numbers (2070 Magh)](https://www.bctengineer.com/question.md?key=semester_8%2Fsimulation_and_modeling%3Apoker-test-numerical-2070-magh) — Simulation and Modeling, chapter 6, asked 3×
- [Poker test, 1000 four-digit numbers (2069)](https://www.bctengineer.com/question.md?key=semester_8%2Fsimulation_and_modeling%3Apoker-test-numerical-2069-bhadra) — Simulation and Modeling, chapter 6, asked 3×

## Related in Simulation and Modeling

- [Poker test, 1000 four-digit numbers (2072 Magh)](https://www.bctengineer.com/question.md?key=semester_8%2Fsimulation_and_modeling%3Apoker-test-numerical-2072-magh) — Simulation and Modeling, chapter 6
- [Poker test, 1000 four-digit numbers (2073)](https://www.bctengineer.com/question.md?key=semester_8%2Fsimulation_and_modeling%3Apoker-test-numerical-2073-magh) — Simulation and Modeling, chapter 6
- [Poker test, 10,000 four-digit values (2070)](https://www.bctengineer.com/question.md?key=semester_8%2Fsimulation_and_modeling%3Apoker-test-numerical-2070-bhadra) — Simulation and Modeling, chapter 6