The first of the two commentaries below is an extended version of my Feb 2000 ENGLISH BRIDGE article ‘Are your hands truly random’ which describes a very simple means of dealing much more random hands at duplicate bridge events. However, in the light of the ensuing constructive comments from John Swingler and other ENGLISH BRIDGE correspondents and the continuing experience at the bridge club at Witney it has also been slightly modified and it now proposes that the rudimentary card-mixing operation be undertaken by the players themselves on the final round of each duplicate session.

The second considers the conclusions of a theoretical study of riffle shuffling, which has been widely reported in the national press and on the Internet as ‘Five shuffles enough for random pack of cards, say scientists’. In particular, it examines the ‘idealisation of a shuffle’, which underlies the mathematical analysis, and it concludes that the actual riffling behaviour of real players is more varied and probably significantly less effective than the study would indicate.

Harry Freeman 

April 2001

This subsequent study is based on a similarly extensive analysis, but with the distinction:

·Firstly, that this second set of duplicate hands were all played at another local bridge club at Witney.

· Secondly, that just before the boards were distributed to the tables to be shuffled and dealt in the usual way the cards were consistently subject to the very simple and rapid pre-mixing procedure described above.

In the Table below the results from this Witney exercise are summarised and they are compared both with those from the quite conventionally dealt hands of the earlier Abingdon study and with the expected card distribution of a similar number of truly random hands.

They support the main conclusion of our earlier analysis that the conventional duplicate card dealing procedure at bridge clubs can be made significantly more random – and the game correspondingly more correct and interesting- by a quite small measure of card pre-mixing before the usual shuffling and dealing of the hands at the beginning of the duplicate event.

THE PREVIOUS ABINGDON STUDY OF OVER THIRTEEN HUNDRED HANDS

In the 1996 BRIDGE Magazine article we noted that the theory and the practice of bridge bidding and play are predicated on the assumption that the cards have a truly randomised distribution.

We concluded that the problem is likely to be particularly marked at those Rubber Bridge games where the tricks –commonly four cards of the same suit – are stacked together at the end of each deal.

But – and this was the main thrust of the article - we also judged that it could adversely affect the card distribution at bridge-club duplicates because of the way in which most players at the end of the final hand simply reinsert their cards in the boards without bothering to mix or sort them.

In consequence, when the packs are reassembled in the usual way for shuffling and dealing at the beginning of the following session they will tend to be made up with sets of hands in which the cards have a similar order[3] .

Despite the fact that such card-ordering is likely to be less pronounced than in rubber bridge we judged that it should still be possible to observe some statistical anomalies in the subsequent deals. And, in addition, we felt that this view was strongly supported by the quite widely held perception among duplicate players that hands which have been dealt at the table in the usual way have less ‘bizarre’ distributions than those which have been generated by computer.

In an attempt to throw more light on this hypothesis we examined in considerable detail the card distribution of over thirteen hundred hands from duplicate events over a period of several months at the bridge club at Abingdon.

The results were fully described in the previous article, but the key factor was the way in which the suits were distributed between the four hands.

The most marked feature – which can be seen clearly in the comparison of the Abingdon hands with the expected distributions in the Table below - was the very significant reduction in the occurrence of voids, singletons and six-card and seven-card suits. But the loss of randomness was also reflected in the way in which the overall distribution of the hands was demonstrably flatter, with a corresponding increase in the probability of the most common suit distributions(note, for example, the 29% increase in the frequency of occurrence of 4-4-3-2 hands!).

The somewhat unexpectedly large scale of these Abingdon discrepancies raised three questions:

1.  Were they relevant to similar duplicate events at other clubs?- We concluded that their detailed nature might vary from club to club, but that their overall effect on the quality of the hands was likely to be similar.

2.If so, would they affect the standard of bidding and play at such club duplicates?– We judged that they would and that this should be a matter of particular concern for bridge clubs, which strive for a high standard of bidding and play. One example - which results from the paucity of six and seven card suits – is the much reduced probability of pre-empts or weak two’s. Another – which is more significant for the play of the hands – is that safety plays to guard against singleton honours become much less worthwhile.

3. How can more random deals be achieved? – It might seem that clubs could aim for a more rigorous shuffling regime when the boards are dealt. But time is usually limited at the start of such events and the reality – as noted above - is that it is surprisingly difficult to arrive at the required degree of mixing. In contrast, however, we went on to note that it would not be difficult for the tournament director to simply ask the players to mix up their hands of thirteen cards before returning them to the boards on the final round. We concluded that this very simple procedure would effectively destroy any symmetry between the four hands and thus ensure that the packs of cards would have a much more random distribution when they were subsequently reassembled for shuffling and dealing.

THE ENSUING ANALYSIS OF OVER THIRTEEN HUNDRED HANDS AT WITNEY

In the light of these Abingdon results – and in particular, the prospect, which they identified of a simple and rapid card-mixing procedure to improve the quality of duplicate-pairs hands – the account below describes a related random-dealing exercise, which has been undertaken at a different local bridge club at Witney. Although Witney is somewhat smaller it has a regular weekly duplicate event, but – and what was particularly important for this study - it closely resembles Abingdon in its consistent use of cover cards, which allowed me to continue the detailed examination of the distribution of the hands.

 
 

Random		Abingdon Card distributions of 1336 duplicate hands dealt normally		Witney Card distributions of 1336 duplicate hands premixed and then dealt normally
Card	Expected No	Observed No	% Deviation	Observed No	% Deviation
Distribution	in 1336 hands	in 1336 hands	from expectation	in 1336 hands	from expectation
voids	68	38	-44	58	-15
singletons	428	350	-18	401	-6
6-card suits	222	165	-26	217	-2
7-card suits	53	38	-29	43	-18`
4-4-3-2	288	372	29	292	1
5-3-3-2	207	228	10	222	7
5-4-3-1	173	161	-7	172	0
5-4-2-2	141	135	-4	138	-2
4-3-3-3	141	150	7	152	8
6-3-2-2	75	57	-24	82	9
6-4-2-1	63	51	-19	56	-11
6-3-3-1	46	34	-26	47	2
5-5-2-1	42	38	-10	42	0
4-4-4-1	40	34	-15	38	-5

CONCLUSION 

As can be seen from the comparison above of these Abingdon and Witney results, this rapid and quite cursory card-mixing procedure has resulted in a quite striking improvement in the random quality of the hands. Moreover, since it has proved to be easy to apply at Witney, initially, as noted above, as a preliminary to the distribution of the duplicate boards, but then increasingly on the final round of the session; and since the perception of the players is that it leads to a more interesting and enjoyable game, it is now used routinely at the club.

A theoretical study of card riffling by two mathematicians published originally in the Proceedings of the Royal Society[4] - and subsequently widely reported in the national press (Daily Telegraph 10^th Oct. 2000) and on the internet as ‘Five shuffles enough for random pack of cards, say scientists’- appears to offer the prospect of a simple solution to the long-standing problem of achieving an adequate degree of card mixing.

In contrast, however, with the newspaper account which simply notes:’that five riffle shuffles – where the pack is split and the cards flapped together at their edges – will make a pack truly random’, the riffle shuffle in the Royal Society analysis is carefully defined in a mathematically precise way. The authors explain that the deck is first cut roughly in half according to a binomial distribution, which they describe. The two halves are then riffled together by dropping cards roughly alternatively from each pile with the probability of a card being dropped from a pile being proportional to the number of cards in it. They go on to note: ‘There is evidence that this idealisation of a shuffle is a reasonable approximation to the actual behaviour of human shufflers’

My concern, which led me to look again at the distribution of packs of cards following such repeated riffling lies not with the mathematical analysis, which is seemingly rigorous and correct, but rather with my belief that the riffling behaviour of real players is more varied and less effective than the study would indicate.

Thus, in the case of experienced players who commonly seek to riffle carefully, rather than roughly, the card-mixing exercise may be surprisingly counterproductive and simply succeed in creating a new degree of order (see Ref 1). A relevant example of the possible outcome of such shuffling can be seen by taking a new pack which is in suit order, dividing it into two equal piles and then reassembling it card by card, as would be the case if it were riffled quite precisely.After five such simulated riffles the reassembled pack may seem at first glance to be reasonably random – AS; 6S;JH; 3H; 8D; KC; 5C; 9S…..But, if it is then dealt in the usual way, the subsequent card distribution is most certainly not so – as can be seen from North’s hand!

S AK7

H AK7

D AK87

And indeed, when I attempted to test out the hypothesis by riffling a number of pre-sorted packs of cards in the prescribed approximate manner, I realised that my own behaviour was not very consistent. For example, I found that it was significantly easier to riffle a part-used deck than a new, or an old one. More importantly, it soon became apparent that the resultant distributions after five rough riffles were not truly random.

I found, for example, that the top card had indeed survived in a number of my riffle shuffle tests. But, additionally, I also found that the overall card separation process had been less effective than I had expected, with the result that in each pack a significant number of pairs of cards were still together.

The exercise was too limited to draw any general conclusions, but at least it convinced me that my own shuffling behaviour is not a good approximation to the mathematical model. On the other hand it was consistent with Andrew Brown’s experience which he described in a 1996 ENGLISH BRIDGE commentary entitled ‘How well do you shuffle? He too found significant numbers of pairs of cards were still together and he concluded that it was necessary to riffle the pack at least ten times to ensure an adequate degree of card mixing

This failure to achieve adequate mixing could be easily detected because we had both used pre-sorted decks with the cards in suit order. It would clearly have been less obvious in the case of real hands at the bridge table. But the real question is whether it would have been significant?

My own judgement, which is based on the extensive analysis of duplicate bridge hands in the first commentary above is that, it would. However, given the many millions of bridge hands, which are shuffled each year in the expectation that the resultant deal will be random, I do believe that the subject merits a more detailed and systematic study before reaching a final conclusion on how many times and how ‘roughly’, to riffle the pack.

In the meanwhile for duplicate bridge events I shall continue to commend the simple card-mixing procedure, described above which does ensure that the hands have a more random distribution.