_
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 |
|
|
|
_
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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