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painfully easy (answer)

"painfully easy  aha:!

i flip a penny and a dime and hide the result from you. "one of the coins came up heads", i announce. what is the chance that the other coin also came up heads? "

The answer should not be 33%, but 50%, because the person doing the coin-flipping said "ONE of the coins came up heads". Implying that the result was 'HT' or 'TH' which is 2 out of 4 possibilities. => 50%

Siddhant Bhansali
Monday, October 6, 2003

Err, if you interpret "one of the coins came up heads" to mean "one and only one of the coins came up heads" then the chances that the other coin also came up heads is, of course zero!

(Incidentally the answer given to the problem is right, before this discussion gets on too far).

David Clayworth
Monday, October 6, 2003

The "most correct" answer would seem to be: "Either 0%, or 50%, depending on the intended meaning of the phrase 'one of the coins'".  (The geek-classic: "It depends...")

As noted, "one and only one" means 0% likelihood, of course.

An intended meaning of "one or more" (inferrable from use of the word "also") yields the "50%" answer.  Barring the use of an un-mentioned, non-assumable three-sided coin, or of the unlikely intent to consider a coin possibly landing on edge, the other coin has two possible states. 

So... will Siddhant Bhansali's correction displace the "answer" currently listed for the question?


Patrick Sweeney
Tuesday, October 7, 2003

No because the answer currently listed is in fact the correct one. Siddhant's solution has been suggested in many threads over the last year or two, but it's not the right one.

Most people have trouble with this because they think that the caller is looking at one specific coin and announcing that it is heads, whereas in fact he is looking at both coins and announcing that at least one of them is heads. Given this it should be clear that only one of the three possible patterns gives both coins heads. (Strictly the wording of the question could have been a little more precise, but that was what was intended).

This one is easy to try out. Throw two coins, ignore any cases where neither coin is heads and see how many of the remainder have both coins heads.

David Clayworth
Tuesday, October 7, 2003

I think that the second event is not related to the first,
so just ignore what happened to the first coin. Second
coin is either one or another side up, therefore it's 0.5

Thursday, October 16, 2003

i agree with the previous post.  the two coins are independent of each other.

Thursday, October 16, 2003

Actually, despite the lack of confidence by the author, the solution given on the solution page is exactly right.  Try it out.

David Clayworth
Monday, October 20, 2003

The solution provided assumes that IF at least one coin is heads, then your host will ALWAYS say "at least one coin is heads." But you don't know this.

What would your host have said if both coins were tails? What conclusions would you make then?

Peter Meilstrup
Thursday, October 23, 2003

It's assumed that the host would throw again, or not pose the question.

David Clayworth
Friday, November 14, 2003

One of the coins came up heads; what is the chance THE OTHER came up heads."

One in two; the other is irrelevant.

"One of the coins came up heads; there is one coin in each hand - what is the chance the coin in my left hand is heads?"

Two in three: there are only three possibilities HH, HT, and TH.

Stephen Jones
Thursday, November 20, 2003

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