---
title: "Poker test, 10,000 five-digit numbers (2074)"
subject: "Simulation and Modeling"
chapter: 6
key: "semester_8/simulation_and_modeling:poker-test-numerical-2074-magh"
source: https://www.bctengineer.com/q/semester_8/simulation_and_modeling/poker-test-numerical-2074-magh
site: BCT Engineering
---

# Poker test, 10,000 five-digit numbers (2074)

Simulation and Modeling · Chapter 6: Random Number

- Asked in: 2074 Magh, 6 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-2074-magh
- 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. Based on Poker Test check whether the numbers are independent. Use χ²(0.05, 6) = 12.592

| Combinations | Observed Frequencies |
|---|---|
| All Different | 3044 |
| One pair | 5020 |
| Two pair | 1090 |
| Three of a kind | 700 |
| Full house | 95 |
| Four of a kind | 40 |
| Five of a kind | 11 |
| Total | 10,000 |

## Short answer

With 7 pattern classes for five‑digit numbers, expected frequencies from standard probabilities yield χ0² = 101.69. Degrees of freedom 6, critical value at α=0.05 is 12.592. Since 101.69 > 12.592, the hypothesis of independence is rejected.

## Answer

**Setup.** H0: the numbers are independent. H1: they are not. N = 10,000, with 7 pattern classes, so df = 7 − 1 = 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 | 3044 | 3024 | 0.132 |
| One pair | 0.504 | 5020 | 5040 | 0.079 |
| Two pair | 0.108 | 1090 | 1080 | 0.093 |
| Three of a kind | 0.072 | 700 | 720 | 0.556 |
| Full house | 0.009 | 95 | 90 | 0.278 |
| Four of a kind | 0.0045 | 40 | 45 | 0.556 |
| Five of a kind | 0.0001 | 11 | 1 | 100.000 |

**χ0² = 101.69**

**Conclusion.** Since 101.69 > 12.592, H0 is **rejected**. The numbers are not independent. The main contribution comes from the "five of a kind" class: 11 were observed against only 1 expected, which alone adds 100.

*Remark:* the other six classes agree closely with the expected counts. If the classes with Ei < 5 are merged (four and five of a kind, Ei = 46, df = 5, critical 11.07), then χ0² = 1.68 and independence would be accepted. But with the critical value 12.592 (df = 6) given in the question, the answer is to reject.

## Check yourself

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

1. With 7 pattern classes, what are the degrees of freedom?
   A. 7
   B. 6
   C. 5
   D. 10000
   Answer: B. df = 7 - 1 = 6, giving critical 12.592. (Key point: Degrees of freedom)

2. For five-digit numbers, P(all different) = 0.3024. What is Ei for N = 10,000?
   A. 302.4
   B. 30,240
   C. 3024
   D. 5040
   Answer: C. Ei = 10,000 x 0.3024 = 3024. (Key point: Expected frequency)

3. Which class contributes most to chi-square0 = 101.69?
   A. All different
   B. Three of a kind
   C. Five of a kind (11 observed vs 1 expected)
   D. Full house
   Answer: C. (11 - 1)^2 / 1 = 100, nearly the whole statistic. (Key point: Main contribution)

4. What is the decision with critical value 12.592?
   A. Accept independence since the other classes fit
   B. Reject independence since 101.69 > 12.592
   C. Accept since 101.69 > 12.592
   D. Reject because N is large
   Answer: B. The statistic exceeds the critical value, so H0 (independence) is rejected. (Key point: Conclusion)

## Close variants set in other exams

- [Poker test, 10,000 five-digit numbers (2072)](https://www.bctengineer.com/question.md?key=semester_8%2Fsimulation_and_modeling%3Apoker-test-numerical-2072-ashwin) — Simulation and Modeling, chapter 6, asked 4×

## Related in Simulation and Modeling

- [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×
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