Another challenge!

The Specific Challenge Asks:

Rakbeater Wrote:You look at your hand and see a 5 card suit with four aces and one ten (AAAAT). No trump has been played and you have the lead. You decide to play your four aces consecutively and follow with the ten unless one of the cards is trumped. The first two aces are winners and not trumped. What percentage of the time will:

the third ace be a winner?

and then the fourth ace?

and then the ten?

Definitions

Acronyms & Terminology

L = Left Hand Opponent

P = Partner

R = Right Hand Opponent

U = Unknown

Y = You

X = Used when specific L, P, R position is not important for discussion

Table Configuration. A specific card distribution in

one suit of one hand dealt to all four Players of the Double Deck Pinochle 80 cards where each player receives 20 cards

Y-L-P-R. When Order Matters. A specific card distribution in

one suit of one hand dealt to all four Players of the Double Deck Pinochle 80 cards where each player receives 20 cards is described with the quadruple Y-L-P-R where order matters. This fully defines the specific Table Configuration for one suit.

An example of Y holding 18 cards of one suit with the remaining 2 cards of the suit distributed to the other three Players, L-P-R, can be fully described by the six configurations, using the notation Y-L-P-R as follows:

- 18-0-0-2

- 18-0-2-0

- 18-2-0-0

- 18-0-1-1

- 18-1-0-1

- 18-1-1-0

Y-X-X-X. When Order Does Not Matter. The configurations are grouped into sets. The configuration is described with the form Y-X-X-X where the order of L-P-R is not important. In the example above with Y holding 18 cards of one suit, the six Table Configurations are grouped into two subsets as follows:

- Group 3 items (1, 2, and 3 from above Y-L-P-R) into 18-2-0-0, and

- Group 3 items (4, 5, and 6 from above Y-L-P-R) into 18-1-1-0.

Y-U. A shorthand notation is employed that describes the Table Configuration as Y-U, where U represents Unknown and sums the other three Player's holdings as U = L + P + R. Thus, for the example above, the Table configuration is defined by 18-2.

Combinations formula. C(n,r) = n! / ( r! • (n-r)! ) where you are choosing r items from n items (or more commonly n choose r)

without regard to order and

without replacement. When order matters then the Permutations formula should be used.

Discussion

There are 136 total Table Configurations for the Y-U configuration of 5-15. These can be summarized into the 27 5-X-X-X configurations listed in Table 1. Column 1 contains an Identification Number, ID, for reference. Column 2 provides specific configurations. Column 3 contains the number of specific configurations of each Y-X-X-X Type. For example, 5-14-1-0 has 6 possible specific configurations: 5-14-1-0, 5-14-0-1, 5-1-14-0, 5-0-14-1, 5-1-0-14, and 5-0-1-14.

Table 1 Summarized Table Configurations for 5-X-X-X

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ID Y-X-X-X Number of Configurations of each Y-X-X-X Type

1 5-15-0-0 3

2 5-14-1-0 6

3 5-13-2-0 6

4 5-13-1-1 3

5 5-12-3-0 6

6 5-12-2-1 6

7 5-11-4-0 6

8 5-11-3-1 6

9 5-11-2-2 3

10 5-10-5-0 6

11 5-10-4-1 6

12 5-10-3-2 6

13 5-9-6-0 6

14 5-9-5-1 6

15 5-9-4-2 6

16 5-9-3-3 3

17 5-8-7-0 6

18 5-8-6-1 6

19 5-8-5-2 6

20 5-8-4-3 6

21 5-7-7-1 3

22 5-7-6-2 6

23 5-7-5-3 6

24 5-7-4-4 3

25 5-6-6-3 3

26 5-6-5-4 6

27 5-5-5-5 1

Check: Counts sum to 136

For those interested, the number of L-P-R configurations given any Y value can be found using C( (n + k - 1), k ) where n = 3 Players (L-P-R) and k = number of cards to distribute among the 3 Players.

The probabilities for each specific configuration are computed using the following methodology.

The Conditional Probability of a specific L-P-R configuration given a specific Y configuration is found by multiplying the Probability of L times Probability of P times Probability of R together.

Pr( L-P-R | Y ) = Pr( L ) • Pr( P ) • Pr( R )

The summation of all individual probabilities for each L-P-R configuration given Y must add to 1 to be a probability distribution.

Σ Pr( L-P-R | Y ) = 1 for all L-P-R configurations given Y

The Probabilities Pr( L ), Pr( P ), and Pr( R ) are found from the following formulas.

Pr( L ) = ( C( (20-Y), L) • C( (60-(20-Y)), (20-L) ) ) / C(80,20)

Pr( P ) = ( C( (20-(Y+L) ), P) • C( (40-(20-(Y+L) ) ), (20-P) ) ) / C(40,20)

Pr( R ) = ( C( (20-(Y+L+P) ), R) • C( (20-(20-(Y+L+P) ) ), (20-R) ) ) / C(20,20)

The probabilities for all 136 possible 5-L-P-R configurations were calculated in a spreadsheet and summarized into the 27 5-X-X-X configurations. Table 2 provides the summarized results.

Table 2 Summarized 5-X-X-X Configurations Probabilities

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ID 5-X-X-X Count Configuration Summed Probability

1 5-15-0-0 3 0.000000000874382862943455

2 5-14-1-0 6 0.0000000874382862943455

3 5-13-2-0 6 0.00000166132743959256

4 5-13-1-1 3 0.00000174876572588691

5 5-12-3-0 6 0.0000161979425360275

6 5-12-2-1 6 0.0000539931417867583

7 5-11-4-0 6 0.0000917883410374892

8 5-11-3-1 6 0.000431945134294067

9 5-11-2-2 3 0.000341956564649469

10 5-10-5-0 6 0.000323094960451962

11 5-10-4-1 6 0.00201934350282476

12 5-10-3-2 6 0.004513826653373

13 5-9-6-0 6 0.000734306728299913

14 5-9-5-1 6 0.00587445382639931

15 5-9-4-2 6 0.0174397847971229

16 5-9-3-3 3 0.0123104363273809

17 5-8-7-0 6 0.00110146009244987

18 5-8-6-1 6 0.0110146009244987

19 5-8-5-2 6 0.0418554835130951

20 5-8-4-3 6 0.0784790315870532

21 5-7-7-1 3 0.00677821595353766

22 5-7-6-2 6 0.0643930515586078

23 5-7-5-3 6 0.154543323740659

24 5-7-4-4 3 0.102626425921531

25 5-6-6-3 3 0.0965895773379117

26 5-6-5-4 6 0.3284045629489

27 5-5-5-5 1 0.0700596400957653

Checks: Counts sum to 136, Probabilities sum to 1.0

Now we can begin to answer Rakbeater's challenge. To be complete all Plays are shown.

First Card

Probability of the 1st card of AAAAT being played and not being Trumped is found from adding the Probabilities of Table 2 that do not contain a 0 in any X position of 5-X-X-X. These are configurations: 4, 6, 8, 9, 11, 12, 14, 15, 16, and 18 through 27. Conversely, one can add all those with a 0 in the configuration and then subtract this from 1.

Probability of 1st card NOT Trumped = 0.9977 = 99.77%

Second Card

Probability of the 2nd card of AAAAT being played and not being Trumped is found from reducing the 27 member set configurations of non-zero 5-X-X-X members by 1 card each. This gives a reduced set of 19 members as shown in Table 3. The ID from Table 1 is maintained for reference.

Table 3 Summarized 4-X-X-X Configurations Probabilities After 1st Card Played

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ID 4-X-X-X Probability (Not Normalized)

4 4-12-0-0 0.000001748765726

6 4-11-1-0 0.000053993141787

8 4-10-2-0 0.000431945134294

9 4-10-1-1 0.000341956564649

11 4-9-3-0 0.002019343502825

12 4-9-2-1 0.004513826653373

14 4-8-4-0 0.005874453826399

15 4-8-3-1 0.017439784797123

16 4-8-2-2 0.012310436327381

18 4-7-5-0 0.011014600924499

19 4-7-4-1 0.041855483513095

20 4-7-3-2 0.078479031587053

21 4-6-6-0 0.006778215953538

22 4-6-5-1 0.064393051558608

23 4-6-4-2 0.154543323740659

24 4-6-3-3 0.102626425921531

25 4-5-5-2 0.096589577337912

26 4-5-4-3 0.328404562948900

27 4-4-4-4 0.070059640095765

Total Cases Remaining = 19 of 27; Probabilities Sum = 0.997731402295116

Probability of the 2nd card of AAAAT being played and not being Trumped is found from adding the Probabilities of Table 3 that do not contain a 0 in any position of 4-X-X-X and dividing by the sum of all of the probabilities in Table 3. The division by the sum of all the Probabilities in Table 3 must be done to normalize the distribution and make it a proper Probability Distribution that adds to 1. These are configurations: 9, 12, 14, 15, 16, 19, 20, and 22 through 27. Conversely, one can add all those with a 0 in the configuration and then subtract this from 1.

Probability of 2nd card NOT Trumped after 1st card played = 0.9738 = 97.38%

Third Card

Probability of the 3rd card of AAAAT being played and not being Trumped is found from reducing Table 3's 19 member set configurations of non-zero members by 1 card each. This gives a reduced set of 12 members as shown in Table 4.

Table 4 Summarized 3-X-X-X Configurations Probabilities After 2nd Card Played

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ID 3-X-X-X Probability (Not Normalized)

9 3-9-0-0 0.000341956564649

12 3-8-1-0 0.004513826653373

15 3-7-2-0 0.017439784797123

16 3-7-1-1 0.012310436327381

19 3-6-3-0 0.041855483513095

20 3-6-2-1 0.078479031587053

22 3-5-4-0 0.064393051558608

23 3-5-3-1 0.154543323740659

24 3-5-2-2 0.102626425921531

25 3-4-4-1 0.096589577337912

26 3-4-3-2 0.328404562948900

27 3-3-3-3 0.070059640095765

Total Cases Remaining = 12 of 27; Sum = 0.971557101046049

Probability of the 3rd card of AAAAT being played and not being Trumped is found from adding the Probabilities of Table 4 that do not contain a 0 in any position of 3-X-X-X and dividing by the sum of all of the probabilities in Table 4. The division by the sum of all the Probabilities in Table 4 must be done to normalize the distribution and make it a proper Probability Distribution that adds to 1. These are configurations: 16, 20, and 23 through 27. Conversely, one can add all those with a 0 in the configuration and then subtract this from 1.

Probability of 3rd card NOT Trumped after 2nd card played = 0.8677 = 86.77%

Fourth Card

Probability of the 4th card of AAAAT being played and not being Trumped is found from reducing Table 4's 12 member set configurations of non-zero members by 1 card each. This gives a reduced set of 7 members as shown in Table 5.

Table 5 Summarized 2-X-X-X Configurations Probabilities After 3rd Card Played

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`ID 2-X-X-X Probability (Not Normalized)`

16 2-6-0-0 0.012310436327381

20 2-5-1-0 0.078479031587053

23 2-4-2-0 0.154543323740659

24 2-4-1-1 0.102626425921531

25 2-3-3-0 0.096589577337912

26 2-3-2-1 0.328404562948900

27 2-2-2-2 0.070059640095765

Total Cases Remaining = 7 of 27; Probabilities Sum = 0.843012997959201

Probability of the 4th card of AAAAT being played and not being Trumped is found from adding the Probabilities of Table 5 that do not contain a 0 in any position of 2-X-X-X and dividing by the sum of all of the probabilities in Table 5. The division by the sum of all the Probabilities in Table 5 must be done to normalize the distribution and make it a proper Probability Distribution that adds to 1. These are configurations: 24, 26, and 27. Conversely, one can add all those with a 0 in the configuration and then subtract this from 1.

Probability of 4th card NOT Trumped after 3rd card played = 0.5944 = 59.44%

Fifth Card

Probability of the 5th card of AAAAT being played and not being Trumped is found from reducing Table 5's 7 member set configurations of non-zero members by 1 card each. This gives a reduced set of 3 members as shown in Table 6.

Table 6 Summarized 1-X-X-X Configurations Probabilities After 4th Card Played

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`ID 1-X-X-X Probability (Not Normalized)`

24 1-3-0-0 0.102626425921531

26 1-2-1-0 0.328404562948900

27 1-1-1-1 0.070059640095765

Total Cases Remaining = 3 of 27; Probabilities Sum = 0.501090628966196

Probability of the 5th card of AAAAT being played and not being Trumped is found from adding the Probabilities of Table 6 that do not contain a 0 in any position of 1-X-X-X and dividing by the sum of all of the probabilities in Table 6. The division by the sum of all the Probabilities in Table 6 must be done to normalize the distribution and make it a proper Probability Distribution that adds to 1. There is only one configuration: 27. Conversely, one can add all those with a 0 in the configuration and then subtract this from 1.

Probability of 5th card NOT Trumped after 4th card played = 0.1398 = 13.98%

Summary

The following probabilities are conditional probabilities given Y holding the AAAAT 5 cards of one suit

after the previous cards are played and not Trumped.

Before 1st card is played

Conditional Probability of 1st card NOT Trumped = 0.9977 = 99.77%

After 1st card is played and NOT Trumped

Conditional Probability of 2nd card NOT Trumped = 0.9738 = 97.38%

After 2nd card is played and NOT Trumped

Conditional Probability of 3rd card NOT Trumped = 0.8677 = 86.77%

After 3rd card is played and NOT Trumped

Conditional Probability of 4th card NOT Trumped = 0.5944 = 59.44%

After 4th card is played and NOT Trumped

Conditional Probability of 5th card NOT Trumped = 0.1398 = 13.98%

Interestingly, if the question is asked as to what the probabilities are

before any cards are played, the conditional probabilities of the 5 cards played and not being Trumped given Y holding AAAAT are then:

Conditional Probability of 1st card NOT Trumped = 0.998 = 99.8%

Conditional Probability of 2nd card NOT Trumped = 0.972 = 97.2%

Conditional Probability of 3rd card NOT Trumped = 0.843 = 84.3%

Conditional Probability of 4th card NOT Trumped = 0.501 = 50.1%

Conditional Probability of 5th card NOT Trumped = 0.07 = 7.0%

The above numbers are all calculated from Table 2 with the methodology shown above WITHOUT normalizing the distributions at each step.

For those interested the probability of AAAAT occurring is:

Pr(AAAAT) = 0.00024 = 0.024%

More understandable (1 / Pr(AAAAT) ) on average 1 in 4,154 hands.

One player at the Table (4 hands dealt per deal) on average will see this 1 in 1,039 deals.

If one assumes an average 8 deals per game then one player at the Table on average will see this 1 in 130 games.

Rakbeater Wrote:The first person to guess the three correct percentages will win a free copy of The Brick Bidding System.

Where's my free book? I still get it even though I didn't "guess", Yes?

RTFQ/RTFA