Re: Me and David C. Ullrich

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?"
>
> 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.
>
> Dr. Ullrich has stated in this forum, that the two mean the same thing.
> If for no other reason than that he said so. Also because everyone
> assumes them to be the same. (everyone does not, I don't)
>

This question has drawn much debate.


Consider the formula:


P(A|B)=[P(B|A)P(A)] / P(B), then, "Two coins were flipped and we were
told 'at least one is a head'".

P(HH|told"at least one H") = (1/4) / P(B)

B is the event which has happened and P(B) is the prior probability
that it happens.

We start with four equally likely outcomes. HH, HT, TH, TT.

P(B) equals the probability that we were told "at least one heads" at
HH, plus the probability that we were told "at least one heads" at HT",
Plus the probability that we were told "at least one heads" at TH, plus
the probability that we were told "at least one heads" at TT.

The answer to our question definitely depends upon these probabilities.
Alter P(B) and get a different answer to our question.

P1. Let P(B) = 1/4 + 0 + 0 + 0 = 1/4 and get answer 1.
P2. Let P(B) = 1/4 + 1/2*1/4 + 1/2*1/4 + 0 = 1/2 and get answer 1/2.
P3. Let P(B) = 1/4 + 1/4 + 1/4 + 0 = 3/4 and get answer 1/3.

P1. P2. and P3. define three different coin flip sequences. In each, we
hear the phrase, "at least one is a head."

Professor Ullrich says that when we hear the phrase "at least one is a
head" we can assume P3. As the three sequences get different answers,
obviously this is an erroneous assumption.

When the numbers in the formula change, the words in the problem
statement have to change also.

Ullrich, and those like him make the erroneous assumptions, I don't. I
get accused of making probability hard. Probability is hard. Making
erroneous assumptions doesn't make it easier.

Eldon:)

.

Relevant Pages

  • Re: OT: The two coin problem, and Wuzzys error.
    ... about their coins, what are the odds that "both of his coins are the ... and it's heads staring back at him. ... If the probability the man says tails when he has one of each is 1, ...
    (rec.gambling.poker)
  • Re: Why online poker is better for the money player.
    ... 27 winners and only 1 of the 9 losers is doubles, ... I also don't know what the odds on heads are, ... watch you toss coins, but sufficiently fast so that he cannot possibly ... >>>anecdotal evidence, basic misconceptions of probability, biased human ...
    (rec.games.backgammon)
  • Re: Me and David C. Ullrich
    ... > language and not in terms of probability as it has nothing to do with ... There are two coins, at least one is a head. ... a tails. ... the hearer knows nothing of heads. ...
    (sci.math)
  • Re: OT: The two coin problem, and Wuzzys error.
    ... about their coins, what are the odds that "both of his coins are the ... and it's heads staring back at him. ... We are left with this result set if, and only if, the probability the ... That would be true, if, and only if, the probability he says "tails" ...
    (rec.gambling.poker)
  • Re: Why the cards must have a memory
    ... Probability theory assumes uncertainty, ... This is why I used the "tiny universe" scenario, ... ***it assumes that the probability is heads half the time FROM NOW ... incident in this case being a coin toss). ...
    (rec.gambling.poker)