This web-page is an expanded version of an article printed in the May 2009 issue of Cribbage World published monthly by the American Cribbage Congress.

The state of Maine deems cribbage as gambling when playing for prize money.   Other jurisdictions, which allow the operation of licensed casinos and lotteries such as Ontario Canada, also have shut down card games played in Legion halls.   They take the view that playing cribbage is gambling, like lotteries.

When you enter a 22-game "cut for deal" tournament you in effect buy a qualifying-lottery ticket where your chances of qualifying will be enhanced or reduced based on how many times you win the cut.   The odds of this lottery can be calculated using probability and confirmed using statistics.

Suppose we have 22 coins and toss them into the air.   There are 23 possible outcomes as seen in the middle of the chart of Cut Wins and Qualifiers.   The probability values for each of these twenty-three outcomes were derived by using a binomial probability function (where n = 22 and p = 0.5), as shown in the top half of the chart.   The bottom half of the chart shows the number of qualifiers from each group.

In cribbage this means that when 1000 players cut for deal in a 22-game qualifying round, each player will fall into one of about 13 groups as shown in the chart.

DeLynn Colvert and others have repeatedly shown that the winner of the cut has a 56% probability of winning.   This is the edge that the cut-lucky groups have.   Some more so than others!

Everyone in each group has the same edge.   Their edge is greater than the groups to the left, and lesser than the groups to the right.   Eight "cut-lucky" players (17 or more cut wins group) will have an edge over the remaining field, 18 players (16 cut wins group) will have an edge over 974 players, 41 players (15 cut wins group) over 933 players, 76 players over 857 cut-unlucky players, and so on!   The further to the right in the column of cut wins, the better the odds of qualifying becomes.   This means that 416 cut-lucky players have an edge in qualifying over the majority of the field, 584 other players.

The numbers in each group for a given tournament will vary somewhat around these numbers (see the resources), but given a large number of repetitions, they will average to these numbers.   If you play a sufficient number of such tournaments your results will be distributed over the groups shown.   However, this does not change the fact that you are participating in a lottery each time, just as if you regularly buy ordinary lottery tickets you will eventually have a winner(s).   However, winning the cut-for-deal lottery does not guarantee winning the tournament or even qualifying, but it does improve the odds of doing so for a minority of the field!

The bottom half of the chart shows the number of qualifiers from each group.   The distribution of the number of wins for each group is again determined by the binomial probability function but with p = 0.56, the probability of a win by dealer and q = 0.44, the probability of a win by initial non-dealer.

What is the effect of having the edge?   For the average player your chances of qualifying in an alternate deal tournament are 1 in 4, exactly 25%.   In the 11 cut wins group (168 players) in a cut for deal tournament 42 or 25% will qualify - exactly the same as in an alternate deal tournament!   We will use 25% as the benchmark.

If you are in the 5 or fewer cut wins group your odds are about 1 in 8 or 13% or about a 50% reduction in your chances of qualifying!   If you are in the 17 or more cut wins group your odds are about 3 in 8 or 38% for a corresponding increase of about 50% in qualifying.   This group's enhanced chances comes at the expense of the 5 or fewer cut wins group.   In fact each of the cut-lucky groups increase in qualifying is at the expense of another group!   Compare the heights of the corresponding group's colors.   Note that there are more qualifiers in the 11 cut-win group than the 12 win group.   This happens because the pool of 11 cut winners has 168 players (25% qualify = 42 qualifiers) whereas the 12 cut winners has a smaller pool of 154 players (26% qualify = 40 qualifiers).

When you pay your entrance fee at Reno in a cut for deal 22-game tournament of a 1000 players, you buy a lottery ticket which will give an edge to 11 of the 23 groups of players as shown in the chart.   However, unlike a normal lottery where you can pick your numbers, you have no say in the matter.   The cut-for-deal process will determine your edge advantage according to the column group in the chart to which you belong.   In principle, it would be exactly the same as if you drew a lottery-ticket from a box of 1000, where each ticket indicates the number of times you are first dealer and then play 22 games with you as first dealer according to the number on your ticket against 22 suitably-matched opponents.

The fact that we do not know who the cut-lucky players will be and that everyone in the field has an equal opportunity of winning this lottery of getting more cut wins then losses, misses the point.   It is still a lottery just as anyone buying an ordinary lottery ticket has an equal chance of being lucky.

Tournament directors should be proactive in adopting the alternate-deal rule to eliminate the lottery that benefits the cut-lucky players - the probability of more wins, skunks, and play-off byes.   Adoption will deprive governments and anti-gambling activists ammunition for painting cribbage as lottery-like gambling.

In 2004, the BOD rejected the proposal to make alternate deal universal because "there was no evidence to support the unfairness of cut-for deal".   I have now provided the evidence that the process is not only inherently unfair but introduces lottery-like gambling into cut-for-deal tournaments.   I propose that the BOD revisit this issue.

RESOURCES.

A demonstration of the top half of the chart can be found at http://ccl.northwestern.edu/netlogo/models/GaltonBox

Interactive sites: http://www.learner.org/courses/mathilluminated/units/7/textbook/03.php

http://edmedia.opb.org:9000/interactive/galton/

DOS simulator jfgalton.zip

The number of wins (0 to 22) can be computed for each group.   Accumulating 13 or more wins determines simplistically the number of potential qualifiers in each group (omitting the effect of skunk game points) resulting in 266 "qualifiers".   Each group is reduced by a factor of 0.0625% to produce the results shown in the bottom half of the chart.

E-mail me if you have any comments or questions.