Re: Me and David C. Ullrich

*** T. Winter wrote:
> In article <1128899117.670195.206830@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> "Elmo" <elmoritz@xxxxxxxxx> writes:
> ...
> > Given at least one heads means that we'll have a success every time at
> > hh, ht, and th. At least one is a head means whatever it says.
>
> No, that is not what is implied. It is implied that the current toss
> has at least one head. Nothing more, nothing less. But you are talking
> about semantics rather than mathematics. You assume that, in some way,
> the sentence "given one is head" means that it would also be possible
> that the sentence would have been "given one is tail". Or in one of it's
> many disguises.
>
I assume that heads and tails are equally likely.
I assume that the problem statement is true.

I know that "two coins were tossed, given at least one head, and given
at least one tail," have the same probability for two of the same.

When two coins were tossed, they can land four equally likely ways. Any
one way could have landed first, therefore we can speak of each of them
individually as though they were the first toss.

Suppose that:
Q1. TT landed first. The statement was generated, "Two coins were
tossed and at least one is a tail. What is the probability for two
tails?" Bruce only heard the statement. He bet one token for TT, and he
will win. What odds should he collect.

Or,

Q2. HH landed first. The statement was generated, "Two coins were
tossed and at least one is a head. What is the probability for two
heads?" Bruce only heard the statement, he bet one token for HH.He will
win, what odds should he collect.

I don't assume, I know that Bruce should collect the same odds at Q1,
or Q2.

Q3. Suppose that at TT, Bob was told "Two coins were tossed and the
dime landed tails. What is the probability for two tails? Bob only
heard the statement, and bet for TT. He will win, should he collect
different odds from Bruce? I don't assume, I know that he should not
collect different odds.

> Good. Let me formulate it differently. Two coins are tossed, and it
> is announced that one is either heads or tails. What is the probability
> that the other is the same? The probability is 1/2.
>
I agree with this scenario.

> Another formulation. Two coins are tossed and it is announced that one
> is heads, or nothing is announced. What is the probability that both are
> equal? Assuming that there is an announcement when at least one head
> crops up, the probability is again, 1/2.
>
I believe that this one is 1/3, but I may not understand exactly what
you're saying.

> A final formulation. Two coins are tossed and it is announced that one
> is heads, or nothing is announced. Assuming that there is an announcement
> when at least one head crops up, what is the probability that, given such
> an announcement, both are the same? 1/3.
I agree with this scenario.

When a scenario is defined, then it is usually not difficult to get a
correct answer. Our job, however is not to describe a scenario, and
therefore a different question which we can answer. Our job is to
answer the question, as written.

We know that the coins were tossed and a statement was made. We have a
one to one ratio of statements to tosses. So long as that ratio stands,
the bettor for two of the same should win half the time. For the bettor
for one of each to win two thirds of the time, the statement should
show a three statements to four tosses ratio.

Example:
Two coins were tossed until at least one is a head. (three statements
out of four, and some kind of demur, pay the winner two to one at HH)
Eldon


> --
> *** t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
> home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~***/

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