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The Yarborough is a hand with no card greater than 9.


635013559600 is the total number of different hands, 347373600 is the number of hands with no card higher than 9. The odds are (635013559600-347373600) to 347373600 against, simplified 2000 to 1 .

What are the odds that you will receive a hand with no card higher than 7 . The number of hands with no card higher than 7 is 2496144 . The odds are 6350111063456 to 2496144 against, simplified 255000 to 1 .

However often the intended question is what are the odds to receive a hand with no card higher than say a 7, but that there is at least one 7 !. This additional restriction results in slightly worse odds…. 77520 is the number of hands with no card higher than 6, so 2496144 – 77520 must be the number of hands where one or more 7s are included but no higher card. That is 2418624 so the odds are 635011140976 to 2418624 against, simplified 263000 to 1 .
To calculate how long you would expect to wait for a specific yardborough type (4 ↔ 9) to occur, a relationship between deals and time must be established. The profile of an enthusiastic player is taken to be 24 boards 4 days a week. Over 50years that is is (96 x 52 x50) =249600 hands.


The number of hands where no card exceeds the indicated rank.

The number of hands that contain at least one card of the indicated rank but no other is higher. These are the differences of the left column

Probabilities. These are the values of the column 2 divided by 635013559600.

The expected number of deals with the highest card equal to the indicated rank occurring over a 50yrs period

Commencing at an arbitrary time point this column is the expected passing of deals before a player sees a hand of the specified type

num DEALS (1/p)

exp WAIT TIME

A

635013559600

442085310304

0.696182

173767.0000

1.4364

1.4364

deals

K

192928249296

141012722864

0.222063

55426.8000

4.5032

4.5032

deals

Q

51915526432

39882303552

0.0628054

15676.2000

15.9222

15.9221

deals

J

12033222880

9722433280

0.0153106

3821.5200

65.3142

2.72143

days

10

2310789600

1963416000

0.00309193

771.7450

323.423

3.36899

weeks

9

347373600

309931440

0.000488071

121.8220

2048.88

5.33

months

8

37442160

34946016

0.0000550319

13.7360

18171.3

3.64

years

7

2496144

2418624

3.80878E-6

0.9507

262551.0

52.5944

years

6

77520

76960

1.21194E-7

0.0303


1.652

millennia

5

560

560

8.81871E-10

0.0002


227.154

millennia

4

0

0

0







total=635013559600

total=1

total=249600




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Some players may never receive a qualifying hand in a 50 yr period. Some more than one.


A probability distribution can be associated to the occasions of yardboroughs. The Poisson distribution is used where events are rare in a large working set of possibilities.
Consider the 7's yardborough and our enthusiast player throughout 50 years - the highlighted line in the table. The Poisson distribution has a parameter ‘mu’ here set to
50/52.59 ~ 1.0, the ratio of 50 years to the expected wait specified in the 7s line.
The probability of NOT being dealt a 7's yardborough in 50 yrs is 0.37. The probability of having only a single 7's yardborough in 50 years is also 0.37. The probability of two occurances
in 50 years is 0.18 and so on.



The extremes are that for some players the day may never come, for others it is the first hand they pick up.

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