One of the allegations is the removal of the tray and placement of the board after the auction is over. The downloadable spreadsheet will let you enter your own opinion of placement of the board, and your own expert opinion of the strongest suit in the hand not on lead. There are calculations in the spreadsheet to work out the probability that this is random. The spreadsheet only covers boards where Fisher/Schwartz sat North/South in the European Bridge Team Championship (EBTC) in 2014.
The specific allegation (from Bridgewinners.com) is:
The partner of the opening leader always controls the tray if they are on defense. If he has no preference, he leaves the tray where it is. If he has a preference, he removes the tray and places the board on the table. The placement of the board signifies his preference as follows:
Spades: The board is pushed toward the opening leader.
Hearts: The board is placed on one side of the table.
Diamonds: The board is placed in the middle of the table.
Clubs: The board is kept on the side of the non opening leader.
Download the spreadsheet by clicking here.
The spreadsheet contains 3 sheets. The first sheet is "Instructions", the second sheet is "Data" to fill out. The third sheet helps generate the links within the spreadsheet and should be ignored. Only fill out values in the "Data" sheet.
It will be easier to explain using hypothetical numbers. We look at the leads where Mr. Fisher/Mr. Schwartz did not take a preference (according to the allegation). Suppose there are 8 boards out of 22 boards that meet this criteria. We look at the boards where an expert decided no preference. Suppose there are 5 boards that meet this criteria.
The mathematical equivalent is an urn containing 22 colored balls. 8 of these balls are blue (no preferences because tray not removed), 14 of them are red (preference). We are going to take 5 balls (boards an expert decided were no preference) out of the urn without replacing any balls. What are the odds that we will select 5 blue balls (no preference) and 0 red balls (preference)?
The problem of selecting balls without replacement from an urn is a well-known mathematical problem. It is known as the hypergeometric distribution problem. Wikipedia has an article on hypergeometric distribution; you can also look on Wikipedia under Urn problem for a description.
If you want to calculate this number yourself, create an Excel spreadsheet as follows. Rows 8, 9, 10 are intermediates values used during the calculation.
| Column\Row | A | B |
|---|---|---|
| 1 | Balls in urn | 22 |
| 2 | Blue balls in urn | 8 |
| 3 | Red balls in urn | =B1-B2 |
| 4 | Number of balls to select | 5 |
| 5 | Blue balls selected | 5 |
| 6 | Red balls selected | =B4-B5 |
| 7 | ||
| 8 | nCr for true positives | =FACT(B2)/(FACT(B2-B5) * FACT(B5)) |
| 9 | nCr for false positives | =FACT(B3)/(FACT(B3-B6) * FACT(B6)) |
| 10 | nCr for total picks | =FACT(B1)/(FACT(B1-B4) * FACT(B4)) |
| 11 | Probability it is random | =B8*B9/B10 |
| 12 | As a percentage | =B11 |
| 13 | As a 1 in XXX number | =1/B11 |
The values in red are formulae and should be entered as shown. Cell B11 should be formatted as a number with several numbers after the decimal point in case of large numbers. Cell B12 should be formatted a percentage. Cell B13 should be formatted as a number. The formulae above will work and you can set any number of balls in an urn.
We ignore the other false positives - boards where Mr. Fisher/Mr. Schwartz did not indicate a preference but an expert did. We would expect there to be some cases because an expert do not know their bidding methods, or their overcall styles, or other information (ability to indicate a lead through doubles/bids etc.) Also if the bidding has indicated a lead then there would be no need to show it through an outside signal - without knowledge of their bidding methods and style the decision on no-preference is more complicated. This paragraph may be misunderstood, so I will try explaining another way.
| Table | Experts | ||
|---|---|---|---|
| No preference | Suit | ||
| Board placement | No preference | Yes | Ignore |
| Suit | Yes | Yes | |
In attempting to disprove this allegation, we are interested in any boards where the alleged board placement for a suit does not correspond to an expert's opinion on the best suit. We are also interested in boards where an expert had no preference, and there was a board placement for a suit. We are not interested, for statistical purposes, in any hands where an expert preferred a suit, but there was a no preference signal according to the board placement. #20 (Monaco board 6) is an example. Holding AKQJT3:QJT8:T8:7 most experts would chose Spades as a preferred suit. In the auction, this player had bid spades and would expect a spade lead. At the table, there was, according to the allegation, no preference shown. These types of hands should not be counted as false positives for logical bridge reasons.
What are the odds of someone looking at the hands and picking the same suit? This is a simple binomial distribution problem which can also be solved using Excel:
We will use an example of 10 hands where there was a suit preference according to the allegation, and 6 of them matched.
| Column\Row | A | B |
| 1 | Hands | 10 |
| 2 | Matches | 6 |
| 3 | Probability each hand | 0.25 |
| 4 | Probability (0..1) | =BINOMDIST(B2,B1,B3,FALSE) |
| 5 | Probability (1 in X) | =1/B4 |
These mathematics are in the spreadsheet.
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