The ACC has been presented with a proposal to replace cutting for deal in sanctioned tournament play with alternate deals. Here are my mathematical arguments in favor of this proposal.
Whoever deals first has about a 10% advantage over his or her opponent. So the process of determining who should deal first should be fair. Mathematics tells us that the process of cutting for deal is inherently unfair in the short run, the world in which we humans experience a finite frequency of events. In the long run a very large number of (in theory, an infinite number) of events is supposedly required for the process to be fair.
First, for the sake of argument, let the odds of winning a single cut be 26/52, ie 50%. In a consolation qualifying round of 8 games, can player X expect a 100% chance of winning 4 cuts? Mathematics tells us that the chances of winning four cuts are 70/256 (see Table I), or from a different perspective that the chances of this NOT happening is 180/256.
This is not fair. Just ask the player who lost all 8 cuts; where the chances are 1/256, far greater than the chances of getting a 29-hand. It will AND does happen. In her next tournament Player Y could have a complete turnaround and win all 8 cuts, or just as likely, lose all 8 cuts again! If Player Y plays long enough (ie an infinte number of times) Player Y will get a 29 hand, win the PowerBall, and win the cut exactly 50% of the time.
Secondly, the process of cutting is unfair in the short run because it removes cards from being cut by Player Y. Showing that the chances of player Y winning a single cut is exactly 50% or not is a challenging mathematical problem.
Let's make some simplifying assumptions: say Player X cuts a 7 and leaves 26 cards available (no 7's left) to Player Y. Three scenarios can occur:
i. the number of available cards higher and lower than a 7 are the same
ii. more lower cards than higher
iii. more higher than lower
In a single cut, suppose it is scenario ii that occurs (12 cards higher). Player Y will have a 12/26 chance, NOT 13/26, of winning this cut, which is not fair.
Note that in the long run, the number of times scenarios ii and iii occurs is equal and the process is supposedly fair but not in the short run, one cut in this case.
Let's examine the long run more closely: 50% of the time there will be an odd number of cards available to Player Y. There are two scenarios:
i. there are none or two cards equal to Player X's card available which means that the number of cards lower is not equal to the number higher (50% in the long run)
ii. there are one or three cards equal to Player X's card available which means the number of cards lower and higher are equal (50% in the long run)
50% of the time there will be an even number of cards available. Again there are two scenarios, but this time
i. there are none or two cards equal to Player X's card available which means that the number of cards lower is equal to the number higher (50% in the long run)
ii. there are one or three cards equal to Player X's card available which means the number of cards lower is not equal to the number higher (50% in the long run)
Even in the long run, there is a 50% chance that Player Y will have a 50% chance of winning the cut. This is not fair. Player Y should have a a 50% chance of winning the cut 100% of the time, not 50%.
Essentially the argument is that when cards are not replaced, cutting for deal is not fair because the composition of the deck is NOT identical for both players in a single cut. Player Y could always have exactly a 50% chance of winning the cut if the same deck composition is available to Y (after X cuts, the top and bottom four cards are removed, the remaining cards are shuffled, then the top and bottom four cards are interchangably restored in place).
Table 1 Game Wins Cut Chance 00000000 1 X0000000 \ 0X000000 | 00X00000 > 8 ... | 0000000X / XX000000 \ X0X00000 | ... | X000000X | 0XX00000 > 28 0X0X0000 | ... | 000000XX / XXX00000 \ XX0X0000 | ... > 56 0000XX0X | 00000XXX / XXXX0000 \ XXX0X000 | ... > 70 000X0XXX | 0000XXXX / XXXXX000 etc 56 XXXXXX00 etc 28 XXXXXXX0 etc 8 XXXXXXXX 1 Total 256
E-mail me if you have any comments or questions.