Re: Me and David C. Ullrich

On 8 Oct 2005 07:12:35 -0700, "Elmo" <elmoritz@xxxxxxxxx> 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?"
>
>Many mathematicians get it confused with "The probability for two
>heads, given at least one head?"
>
>I say that "given at least one head", and told, "at least one head"
>mean two different things.
>
> ...
>
>When you accept that "given at least one" and told, "at least one" are
>different; then it's easy to get from:
>
>P(hh|at least one h) = 1/3 to
>P(hh|told "at least one h) = 1/2.
>

If information is "given", how is it given? Someone tells you, right?
I can't think of any other way to interpret "given".

For example, consider the following 2 problems:

(1) Given x=3. What is the value of x+1?.

(2) You are told x=3. What is the value of x+1?

I'm certain that most people would not see any difference in the above
problems.

As another example, let's take your example but with just 1 coin.

Consider the following 2 problems:

(1) One coin is flipped. Given that there is at least one head, what
is the probability that it is a head?

(2) One coin is flipped. If you are told that there is at least one
head, what is the probability that it is a head?

It's obvious to you, I hope, that the answer to both questions is 1.
Ok, so again, no difference.

Now for the problem you raised with 2 coins, you seem to think that
being "given" some info and being "told" the same info is somehow
different, and as a result, we get 2 different answers for the
following problem:

2 coins are flipped. You are told that there is at least one head.
What is the probability of 2 heads

You think the answer is 1/2 whereas my answer is 1/3.

To bring you back to reality, I'll propose a bet based on the
following experiment.

A referee flips 2 coins, but flipped coins are hidden from public
view. If both are tails, the referee reflips both coins. If at least
one is a head then the referee announces:

"There is at least one head."

If you think the probability of 2 heads is 1/2, then presumably you
would be willing to bet even money on 2 heads. After all, you view it
as 50-50, right?

Well, if you really believe it's 50-50 then quick, go get some money,
as much as you can afford to lose, and let's go get a ref and a fair
coin. You bet on 2 heads, I'll bet against.

To make the bet more attractive, instead of even money, I'll offer you
3-2 odds. In other words, if the result is 2 heads, you win and I'll
pay you 3, otherwise you lose, but you only have to pay me 2.

If you agree to play this game, one of us is sure to go broke. Who do
you think that would be?

quasi
.

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