[inquiry]

Ok, in a bridge related thread I came up with a crazy number of 1 in more than a quarter billion chance for an occurrence. I have been told that my method was horribly wrong and the correct number is one in 6.9 million. Quite a difference.

Here is the hypothetical position being tested (any similarity to real world is probably more than a coincidence).... There are 28 doors, behind 9 of them is a million dollar prize you get to keep, behind 19 of them is a monster who will eat you if you open that door. There is no indication of which doors have a prize and which has a monster. You must open nine doors. What is the odds you will end up alive with 9 million and alive?

Your first door, your chance of finding a million is 9 out of 28 (9/28).
Your second door, your chance of finding a million is 8 of 27 (8/27),
etc until if you are still alive as you open the last of your nine doors the odds are 1 out of 20.

The combined chances, I have been told, is 9/28 x 8/27 x 7/26 ..... to 1/20 = 362880 out of 2,506,380,000,000 or simplified to 1 out of 6,906,900

This model is, of course, based on the 14 deals (individual 28 hands) on the youtube video from Bali. Out of the 28 North-South hands, nine hands had shortness. Without any concern about the meaning of a cough, it can be observed that on all nine hands with shortness the player with shortness coughed during auction, but not once on the 19 hands The problem with the house/door/monster/prize analogy of course is you know prior to the puzzle that you only have to open 9 doors. In the case of the bridge hands there will be an unknown number of hands so there might be need to be extra weighting to non-cough.

Anyway, anyone know the way to approach the simplified version of what the probability is that coughing in the example above? Is such calculations anyway comparable to similar calculations where examining coughing occurs by accident on 9 out of 9 hands with shortness and no coughing occurs on the 19 out of 19 hands with no shortness.

[awm]

The problem with the monsters would seem to be correctly solved.

However, in the original problem it seems unlikely that they are selecting exactly nine of twenty-eight boards at random on which to cough. Instead, it makes more sense that (if they really weren't cheating) there would be some fixed identical probability of coughing on any board. If this probability is 9/28 (maximizes their chances), then the chance of coughing on exactly the nine boards with shortness would be (9/28)^9 * (19/28)^19 = about one in 43,243,802.

[ArtK78]

Also, there is the chance (however small that it may be) that the cough was an involuntary response to external stimuli rather than a signal.

After all, I sometimes cough during the auction or prior to the opening lead, and it has nothing to do with my distribution or the suit that I want led.

[mike777]

Is this a debate about stats?

If only bridge ok....nevermind

If about macro stuff non bridge tuff?

[Mbodell]

awm, on 2014-April-19, 22:09, said:

The problem with the monsters would seem to be correctly solved.

However, in the original problem it seems unlikely that they are selecting exactly nine of twenty-eight boards at random on which to cough. Instead, it makes more sense that (if they really weren't cheating) there would be some fixed identical probability of coughing on any board. If this probability is 9/28 (maximizes their chances), then the chance of coughing on exactly the nine boards with shortness would be (9/28)^9 * (19/28)^19 = about one in 43,243,802.

In the reality you'd expect there to be some probability of coughing given a period of time, not just measured by hand. Maybe you wouldn't expect a cough to follow closely after another cough, but you'd expect coughing to occur at some sort of rate (probably approximating a Poisson process). The key bit I'm getting at is you'd expect a reasonable number of the coughs to take place during the play, especially since the play was more of the time than the auction. That the coughs tracked so closely with auction and opening leads make it even more unlikely to be random than the math above.

The math for the monsters is just 1 in 28 choose 9 which is indeed 6,906,900 according to excel.

[Trinidad]

One needs to separate statistics into two areas:
- Those needed to propose a model
- Those to verify a model (test a hypothesis)

In the first case, data are gathered to see whether different parameters are correlated. This is sometimes called data mining. When a correlation is found, a hypothesis is postulated describing the relation between the parameters.

This is what Eddie Wold did in the third segment: He gathered data on coughs and compared them to the hand records. He found two correlations. And he postulated two hypotheses:
1) The number of coughs at the start of the auction indicates shortness: 0 coughs for no shortness, 1 for ♣, etc.
2) The number of coughs when partner is on opening lead, at the end of the auction indicates a desire for an opening lead: 0 coughs for no desire, 1 for clubs, etc.

An independent researcher (WBF) set out to test the hypotheses. For that he needs a new, separate set of data. This is the fifth segment, and is shown on the videos. This means that there was a hypothesis that could be tested. We are no longer looking for coughs in the auction as compared to whether E-W were coughing during the coffee breaks or during the play. We are going to observe the coughing pattern and predict the shortness situation (hypothesis 1) or the desired opening lead (hypothesis 2).

It is important to note that to do that:
- the independent researcher cannot know the data that he needs to predict. He cannot know about the hand records, and he must make his observations before he knows what the actual opening lead is.
- the independent researcher does need to know the model.
- we are not comparing observed coughs to distributions. That would be comparing apples and oranges. We are comparing predicted distributions (from the code) to real life distributions (from the hand records).

So, ideally the independent researcher has a sheet in front of him:

     Before auction   pred. shortness    After auction    pred. desired lead
Board       E  W           E    W                E    W   
      1        2   0            ♦    BAL              0    1                   ♣

After the session, he will compare the two sets of predictions before the fact, and reality after the fact:
1) Did the predicted shortness situation match the shortness in the hand records? (Yes/No)
2) Did the predicted desired opening lead match the actual desired opening lead? (Yes/No) (This is a little difficult, since it requires bridge judgement to pick the actual desired opening lead.)

Ben did test a slightly different hypothesis
3) Does coughing (Yes/No, irrespective of the number of coughs) predict shortness (Yes/No, irrespective of the location of shortness)?

He found that Coughing (Yes/No) predicted Shortness (Yes/No) in all cases. And now he tried to calculate what the probability was that this was a coincidence. In his calculation, he made a common error.

He tried to calculate based on:

- A hand either contains shortness, or it does not. (Correct)
- Since the answer is either Yes or No, the probability for each is 50%. (Oops! Not correct.)

We can understand that we need to involve the probability for shortness in the calculation if we try to predict something where the probability is far away from 50%. I will to predict the next 100 hands for Ben. My hypothesis is:

"If Ben says something understandable in Maori then he will hold 13 clubs. If he doesn't say anything understandable in Maori, he will not hold 13 clubs."

I am pretty sure that this model will be able to "predict" whether Ben holds 13 clubs in the next 100 hands.

Following Ben's reasoning, you either hold 13 clubs or you don't, so each has a probability of 50%. Therefore the probability for this being a coincidence will be 1/2¹⁰⁰. We will accuse Ben of conveying to his partner whether he does or does not hold 13 clubs and he will be suspended.

So, Ben's analysis is "halfway correct" (i.e. wrong ) He will have to correct for the probability that a hand does or does not contain shortness. In a single hand, to a bridge player, this seems to be about 50-50, so it is hard to catch Ben's error (and much easier to catch the error in the Maori-club relation). In reality, the probability for any shortness in a hand (and, hence, a cough by E./W.) is 35.7%. Close enough on a single hand, but when multiplied over several hands, these errors are adding up.

Rik

[PrecisionL]

Well done.

My spread sheet gives P(at least 1 singleton) = 30.79 % and P(at least one void) = 5.11 %.

However, P(either shortness) = 35.66% [Several distributions have a singleton and a void]

[hrothgar]

Mbodell, on 2014-April-20, 02:16, said:

In the reality you'd expect there to be some probability of coughing given a period of time, not just measured by hand. Maybe you wouldn't expect a cough to follow closely after another cough, but you'd expect coughing to occur at some sort of rate (probably approximating a Poisson process). The key bit I'm getting at is you'd expect a reasonable number of the coughs to take place during the play, especially since the play was more of the time than the auction. That the coughs tracked so closely with auction and opening leads make it even more unlikely to be random than the math above.

Mbodell has raised a really important point here...

The likelihood that the timing of the coughs would correspond to times where its valuable to exchange information is equally significant.