Re: Me and David C. Ullrich


john_ramsden@xxxxxxxxxxxxxx wrote:
> john_ramsden@xxxxxxxxxxxxxx wrote:
> >
> > [...]
> >
> > So although, a priori, "at least one head" and/or
> > "at least one tail" are symmetric, among all the
> > (unordered) possibilities TT, HT, HH, once the two
> > coins have been thrown, and we are told one or the
> > other "at least", then only _two_ possibilities can
> > apply a posteriori:
> >
> > * TT or HT for "at least one tail"
> > (in which HT has _incidently_ "at least one head")
> >
> > * HT or HH for "at least one head"
> > (in which HT has _incidently_ "at least one tail")
> >
> > I have a feeling others have made this point more
> > or less explicitly earlier in the thread; but it
> > bears repeating (about ten times!).
>
> Just one more time (second or third?), slightly
> more explicitly:
>
> The guy throws two coins (together - we're not
> concerned with the order here, thank God, or it
> would be even more complicated!)
>
> If the coins land either TT or TH, he can then
> pose the problem "I threw two coins, and at least
> one is a tail. What is the probability they are
> both tails?".
>
> If the coins land either TH or HH, he can then
> pose the problem "I threw two coins, and at least
> one is a head. What is the probability they are
> both heads?".
>
> Note that if the coins land TH then the guy can
> pose either problem. But even so, he decides,
> by whatever means, on one problem to ask you.
> In this case it doesn't matter how he decides -
> Probability is only ever relative to a given
> state of knowledge, and barring any further
> information on his decision process you must
> either assume it is random or else that the
> answer to his question is, for you, unknowable!
>
> Surely you can see now that, although "up front"
> (a priori) the coins can land in any of three
> possible ways, for whichever question the guy
> actually asks after the throw (a posteriori)
> there are only _two_ equal possibilities, and
> thus the answer to the problem is 1/2.

When two coins are tossed, they land one of four equally likely ways.

We have the statement, "Two coins were tossed and at least one landed
heads." We therefore know that "heads" were chosen. Were tossed is past
tense, we know we are talking about a historical event, an event that
has already happened.

As Heads and Tails started equally likely, they were equally likely to
be chosen. "Tails" were equally likely to be chosen, up to and until
"heads" were chosen.

The answer is a function of when the choosing was done. When the
choosing was prior to the inspection, then the answer is 1/3. When the
choosing is after the inspection, when at HT, and TH either was still
equally likely, then the answer is 1/2. Ullrich can't assume it to be
one third.

Eldon

.

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