Initial Layout Probabilities

PROBABILITIES OF DIFFERENT COMBINATIONS OF CARDS SHOWING IN THE INITIAL LAYOUT FOR KLONDIKE SOLITAIRE

B. A. Hughes

2017 November 16

Patterns can occur in the 7 cards that are turned up in the initial layout for solitaire. Some of these patterns can seem to appear too often when comparing one layout to another and that can lead to questions about the randomness of the cards. As examples of patterns, the above image shows 2 two-rank straights (3-4 and 10-J), 5 cards of one colour (red), a maximum of 3 cards in one suit (diamonds), and one pair (10s). The frequencies with which four common patterns should show up, on average, are as follows:

Six cards of one colour....once in 11 layouts
Three-of-a-kind................once in 20 layouts
Five-card straight.............once in 28 layouts
Five-card flush.................once in 35 layouts

Probabilities and frequencies of occurrence of all the multiples (pairs, full houses,..) and all the straights, flushes, and colours that are possible are given in the following 4 tables:

In the extra cards flushes and straights are ignored

Multiples Within Ranks	Frequency	Probability	Average Number of Layouts to a Repeat
All cards with different ranks (P₁)	28,114,944	0.210150887	4.8
One pair (P₂)	63,258,624	0.472839497	2.1*
Two pairs (P₂₂)	29,652,480	0.221643514	4.5
Three pairs (P₂₂₂)	2,471,040	0.018470293	63
One 3-of-a-kind (triple) (P₃)	6,589,440	0.049254114	20*
One 3-of-a-kind & a pair (P₃₂)	3,294,720	0.024627057	41
One 3-of-a-kind & 2 pairs (P₃₂₂)	123,552	0.000923515	1,082
Two 3’s-of-a-kind (P₃₃)	54,912	0.000410451	2,436
One 4-of-a-kind (quadruple) (P₄)	183,040	0.001368170	731*
One 4-of-a-kind & one pair (P₄₂)	41,184	0.000307838	3,249
One 4-of-a-kind & one 3-of-a-kind	624	0.000004664	214,398
TOTAL (52C7)	133,784,560	1

* 1.4 for a pair alone or combined with other pairs; 13 for a triple alone or combined with other triples; 595 for a quadruple alone or combined with other quadruples.

Aces are always low, flushes are ignored, and pairs (etc) in any of the cards are ignored
Multiple straights count only as the highest one, and 2 or more that are equally high count as one

Straights	Frequency	Probability	Average Number of Layouts to a Repeat
all cards from non-adjacent ranks	6,471,184	0.04837019	21
2-rank straights	59,034,848	0.44126802	2.3
3-rank straights	45,663,744	0.34132298	3.9
4-rank straights	16,790,784	0.12550614	8.0
5-rank straights	4,726,272	0.03532748	28
6-rank straights	983,040	0.00734793	136
7-rank straights	114,688	0.00085726	1,167
TOTAL (52C7)	133,784,560	1

Multiples and straights are ignored

Colours	Frequency	Probability	Average Number of Layouts to a Repeat
4 cards of one colour	77,740,000	0.581083497	1.7
5 cards of one colour	42,757,000	0.319595924	3.1
6 cards of one colour	11,971,960	0.089486859	11
7 cards of one colour	1,315,600	0.009833721	102
TOTAL (52C7)	133,784,560	1

Multiples and straights are ignored

Flushes	Frequency	Probability	Average Number of Layouts to a Repeat
maximum of 2 cards in one suit	24,676,704	0.184451061	5.4
maximum of 3 cards in one suit	78,881,088	0.589612792	1.7
4 cards in one suit	26,137,540	0.195370378	5.1
5 cards in one suit	3,814,668	0.028513515	35
6 cards in one suit	267,696	0.002000948	500
7 cards in one suit	6,864	0.000051306	19,489
TOTAL (52C7)	133,784,560	1

There are comparable probabilities available in the public domain which show identical values to these for identical cases. In particular, the probabilities for some 7-card poker hands are directly obtainable from probabilities for rank multiples, flush and straight:

Item	Present Analysis	Present Analysis Frequency	Poker Hand Frequency
Full House	P₃₂ + P₃₂₂ + P₃₃	3,473,184	3,473,184
4-of-a-Kind	P₄ + P₄₂ + P₄₃	224,848	224,848
5-card Flush	P_F₅ + P_F₆ + P_F₇	4,089,228	4,089,228*
5-card Straight	P_S₅ + P_S₆ + P_S₇	5,824,000	5,824,000^†

* includes straight flush and royal flush — the present probabilities include A,2,3,4,5
and 10,J,Q,K,A because rank is not important in the present calculation of flush
probabilities, however both of these are part of the straight flushes that have not
been included in the flush probabilities for poker (4,047,644) so they both have to
be added back in (37,260 is the straight flush count for Ace up to King, and 4,324
is the count for royal flushes).

^† from Alspach, with modifications so flushes are not removed and Ace is only low.

THE COMPLETE ARTICLE with derivations of the probabilities,
references, and a comparison with the layout results
from a computer solitaire game.

A. Introduction

In the initial layout of Klondike Solitaire there are patterns in the 7 turned-up cards that can be interpreted as an indication of ineffective randomization of the cards before the layout, or of some artifact leading to unexpected correlations occurring in the order of the cards. In order to provide a basis for deciding whether this interpretation is correct or not, the present analysis determines the mathematical frequencies of a variety of patterns assuming no inherent order before the layouts. The patterns chosen for analysis are in 4 categories and are the following: 11 kinds of multiples within ranks (pairs, threes-of-a-kind,…), 7 kinds of straight distributions, 4 kinds of colour distributions, and 6 kinds of flush distributions — in all, a total of 28 patterns. In the mathematical analysis, it will be necessary to use the total number of ways that 7 cards can be chosen from 52 without regard for their order, which is 52C7 (133,784,560), where nCm is the binomial coefficient m!/[n!(m-n)!]. For each of the four categories the number of kinds make a complete set, so that the sum of probabilities of the kinds in each category is unity, and the total number of ways all the kinds of patterns in a given category…..(more)