Poker's Test for Independence of Random Numbers| Example for Poker's Test

This test is named after a game of cards called poker. In this game five cards are distributed to each player out of pack of fifty two cards. The cards are ranked from high to low in the following order:

Ace, King, Queen, Jack, 10, 9, 8, 7, 6, 5, 4, 3, 2 

Aces are always high. Aces are worth more than Kings which are worth more than Queens which are worth more than Jack, and so on. Each player is dealt five cards. The object of the game is to end up with the highest-valued hand. From best to worst, hands are ranked in the following order:

1. Royal Flush: A Royal Flush is composed of 10, Jack, Queen, King and Ace, all of the same suit.
2. Straight Flush: Comprised of five cards in numerical order, all of the same suit.
3. Four of a Kind: Four cards of the same numerical rank and another random card.
4. Full House: Of the five cards in one’s hand, three have the same numerical rank, and the two remaining card also have the same numerical rank.
5. Flush: A Flush is comprised of five cards of the same suit, regardless of their numerical rank. In a tie, whoever has the highest ranking card wins.
6. Straight: Five cards in numerical order, regardless of their suits.
7. Three of a Kind: Three cards of the same numerical rank, and two random cards that are not a pair.
8. Two Pair: Two sets of pairs, and another random card.
9. One Pair: One pair and three random cards.

Poker test not only tests the randomness of the sequence of numbers, but also the digits comprising of each number. Every random number of five digits or every sequence of five digits is treated as
a poker hand. For example-

13586 are five different digits (Flush)
44138 would be a pair
22779 would be two pairs
33342 would be three of a kind
55533 would be full house
55556 would be four of a kind
77777 would be five of a kind

The occurrence of five of a kind is rare. The order of 'cards' in a 'hand' is unimportant in the test.

Example: Poker Test

A sequence of 1000 four-digit numbers has been generated and an analysis indicates the following  combinations & frequencies:

Combination Distribution (i)	Observed Frequency (Oi)
4 different digits	540
3 like digits	50
4 like digits	20
1 pair	320
2 pairs	70

Use Poker’s test to determine if these random numbers are independent, α = 0.05.

Solution:

In 4-digit numbers, there are only five possibilities- four different, 3 of a kind, 4 of a kind, 1 pair and 2 pairs. Let’s calculate the probabilities for each of these cases:

Case-I: P(4 different digits)

= P(2^nd diff. from 1^st) * P(3^rd diff. from 1^st & 2^nd) * P(4th diff. from 1^st, 2^nd & 3^rd)

= 0.9 * 0.8 * 0.7 = 0.504

Case-II: P(3 like digits)

            = P(2^nd digit same as 1^st) * P(3^rd digit same as 1^st) * P(4^th digit diff. from 1^st) * ⁴C₃

            = 0.1 * 0.1 * 0.9  * 4= 0.036

Case-III: P(4 like digits)

            = P(2^nd digit same as 1^st) * P(3^rd digit same as 1^st) * P(4^th digit same as 1^st)

            = 0.1 * 0.1 * 0.1 = 0.001

Case-IV: P(1 pair)

= No. of possible combinations for a pair from 4 digits * P(2^nd digit same as 1^st in the pair) * P(3^rd digit diff. from 1^st) * P(4^th digit diff. from 1^st & 3^rd)

= ⁴C₂ * 0.1 * 0.9 * 0.8 = 0.432

Case-V: P(2 pairs)

            = 1 - P(4 different digits) - P(3 like digits) - P(4 like digits) - P(1 pair)

            = 1 – 0.504 – 0.036 – 0.001 – 0.432 = 0.027

With N = 1000, let’s summarize the results for Poker’s test in the following table:

Combination Distribution (i)	Observed Frequency               Oi	Expected Frequency           Ei = Prob.*N	(Oi – Ei)2/Ei
4 different digits	540	504	2.571
3 like digits	50	36	5.44
4 like digits	20	1	361
1 pair	320	432	29.037
2 pairs	70	27	68.48
	1000	1000	466.528

Calculated χ² value = 466.528

Degree of freedom = n – 1 = 5 – 1 = 4

At α = 0.05, acceptable value of χ² from table = 9.49

Since, the calculated value of χ² is greater than the acceptable value at four degree of freedom, the independence of the random numbers is rejected on the basis of this test.

Download the Solution (.pdf)

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Enjoy testing random numbers !!!!!