han, on 2011-September-02, 07:30, said:
Consider a bag with 2 red balls and 2 white balls. Suppose that our RHO randomly takes two of the 4 balls out of this bag, and we learn that he must have at least 1 red ball. What is his expected number of white balls?
Using your empty space method, you would get 2/3. After all, he has 1 "empty space", and 2 of the "remaining" 3 balls.
If we number the red balls A and B, and the white balls C and D, then we see that there is 1 way to draw two red balls (A+B) and there are 4 ways to draw a red ball and a white ball (AC, AD, BC, BD). Hence of the 5 possible drawings with at least one red ball, 4 have one white ball and 1 has no white ball, for an average of 4/5 white balls.
Where is your mistake? The answer you get would be correct if you knew that RHO had a specific red ball. For example, if we knew that RHO has ball A, then the expected 2/3 white balls is correct. But we don't know this, we know he has at least 1 red ball, a very different statement.
So what's the exact expected number of diamonds for RHO? I don't know, it is a very difficult computation. To get the right answer we would need to take HCP into consideration as well, which makes the question impossible to answer without knowing RHO's opening tendecies. Rounded to the nearest integer I am pretty sure that the answer is still 3 though. I don't know why this is actually interesting, but when somebody starts posting incorrect mathematics I feel a strong urge to correct them before anybody gets harmed.
Thanks for the correction
Without detracting from that thanks, which I mean, and fully (I think) understanding Helene's additional point about integers, the purpose of the exercise is not to calculate with precision how many cards any player 'holds' in a suit, but, rather, to estimate the likelihood that we will find adequate or inadequate support. Thus is doesn't matter, at all in the context of at the table bridge, as opposed to being presented as a math exercise, if the calculation would reveal the expectation to be 3.0762 or 2.9263. We can live with '3'.
If it were 2.49, then we'd expect '2-3' and if it were 3.49 we'd expect '3-4'. while knowing all along that it could be 0 or (f we held 5 in the suit) 8 or anywhere inbetween.
So as long as my incorrect math will give me something that is going to be within a workable margin of error, I'll keep doing it, while never elevating the result beyond an estimate.
Btw, I do understand and empathize with the urge to correct 'obvious' (obvious, that is, to you...not, as it happened, to me!) errors such as the ones I made. When I see television or movies about lawyers, I often get irritated by actions or dialogues attributed to lawyers that would never happen. So when those of us who are mathematically literate see a post by someone such as me in which it is clear I don't understand the topic I am spouting off on, I should and do expect correction, and I appreciate it.