Re: Me and David C. Ullrich
- From: quasi <quasi@xxxxxxxx>
- Date: Thu, 13 Oct 2005 18:51:19 -0700
On 13 Oct 2005 14:24:16 -0700, sugnaboris@xxxxxxxxx wrote:
>
>Elmo wrote:
>> sugnaboris@xxxxxxxxx wrote:
>> > Elmo wrote:
>> > > sugnaboris@xxxxxxxxx wrote:
>> > > > Elmo wrote:
>> > > > > On this forum I have argued the question, "Two coins were flipped and
>> > > > > at least one is a head. What are the chances that there are two heads?"
>> >
>> > <snip>
>> >
>> > > > If you accept that this meets your criteria, then it will be very easy
>> > > > to settle the discussion with a real-coin or computerized simulation;
>> > > > if you don't accept it, could you please propose your own experiment in
>> > >
>> > > > equivalent terms?
>> > > If you will read a little further, we are examining the difference
>> > > between, "Two coins were flipped and at least one is a head", and "Two
>> > > coins were flipped, given that there is at least one head." I will
>> > > accept your program for "given at least one head." It's easy to see
>> > > that it will give 1/3.
>> >
>> > <snip>
>> >
>> > Yes, but the point of my proposing an experiment, then asking you to
>> > propose your own experiment in equivalent terms, was to give you an
>> > opportunity to define *exactly* what you mean in unambiguous terms that
>> > can be checked experimentally. If you can't describe an experiment in
>> > this way, then it seems to me that you have no proper definition of
>> > what you mean.
>> >
>> > Can you describe the experiment that would give a probability of 1/2 in
>> > the same sort of detail that I gave for the 1/3 case? If you can, I am
>> > happy to run a simulation for you. However, as you are no doubt aware,
>> > setting up an experiment of this sort requires all steps to be
>> > described completely unambiguously, and in an imperaitve/procedural
>> > fashion.
>> >
>> > Look forward to seeing your experiment proposal.
>>
>> We need two computer programs.
>> Call them P1 and P2.
>>
>> P1 randomizes the coins, and color codes the outcomes.
>> P1 passes the color code to P2.
>> P1 shows us two colors.
>> We see two colors, for example, red and green.
>> We can say, "at least one is red", or "at least one is green".
>> P2 has the color code, takes our "at least one is" choice, and
>> announces the outcome, in terms of, "at least one is a head", or "at
>> least one is a tail".
>>
>> Eldon
>
>It sounds like you aren't able to specify an experiment that could be
>carried out that would give the probability of 1/2. You can forget all
>about the details of the implementation, because that's completely
>trivial (OK, RNG isn't trivial - but all we're trying to do is
>discriminate between 1/2 & 1/3). It makes no odds as to whether two
>programs are used, really it doesn't. Generating the trials and
>checking them can be done independently of each other, within a single
>program, just by separating the concerns.
>
>So, again: are you able to write down a set of logical steps such as
>"flip this number of pairs, discard these pairs, count these
>occurences, divide this number by that number"? If you can do that,
>it's very easy to test your hypothesis; but if you can't identify how
>to gather the data for such a simple test, then it's very likely that
>you don't have a concept that really represents a probability. All we
>need is for you to identify the "successful" outcomes, then we can run
>a large number of trials, and divide the successful ones by the total
>(as I described in my example experiment, a few posts ago).
>
>Would you like to define a proper experiment, or do you think that
>there is no real experiment that would give the result that you claim
>(or is there a 3rd option)?
In a recent post in this thread, he did specify an experiment which
forces the probability of 1/2, namely this one:
A referee flips 2 coins.
Based on the results (visible only to the ref), the ref makes one of
the following 2 announcements:
(1) there is a least one head
(2) there is at least one tail
In a case where either statement is possible, the ref picks one of the
2 announcements at random (equally likely).
Based on the above experiment, if the ref announces "there is at least
one head", then the probability of hh is clearly 1/2, not 1/3.
So the issue here is not a mathematical one, but one of language.
I discussed this in a prior reply in this thread.
If all you are told is "there is at least one head", and if the
problem never mentions that there were other possible announcements,
then you can't invent these other scenarios.
You have no information in the given problem saying that the
announcement "there is at least one tail" was even possible. The
problem just states that you were told "there is at least one head".
If alternative announcements were possible, then the problem is
obligated to specify them and also the rules by which they would be
chosen, since such alternatives clearly have the potential to affect
the answer. Since the problem makes no such declarations, there's no
reason to assume any other statements were possible.
If the only possible statement the ref could make was "there is at
least one head", and if the ref is required to make that statement
whenever it can be said, then if you are told "there is at least one
head", the probability of hh is 1/3.
quasi
.
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