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'painfully easy' actually undefined.

This is one of those semantic language problems. The question is not clear enough to answer certainly.

If you consider the sentence "One of the coins came up heads" as a statement about only one coin, leaving the state of the other coin undefined, then the answer is 50%, since any coin can come up heads or tails.

If you consider the sentence to be a summary report on both coins, then the answer is 0%, since you were just told that only one of the coins turned up heads.

Unfortunately, there is no way to disambiguate the statement. Therefore, the "puzzle" isn't really definitively solvable.

Mike Weber
Saturday, June 15, 2002

Consider the statement to be "at least one of the coins came up heads."  This removes the ambiguity.

Roy Pollock
Wednesday, June 19, 2002

Alas, even that doesn't remove the ambiguity. To see why, try to reproduce the experiment 100 times.

Each time, you flip two coins. About 1/4 of the time, they both come up heads. About 1/2 of the time, they come up mixed. About 1/4 of the time, they both come up tails.

Clearly, when they both come up tails, you don't announce that one came up heads. So, sometimes you do something different. You might, for example, say "oops" and reflip the coins. Or maybe, for a different example, you might announce that one came up tails.

The ambiguity comes from the fact that the flipper's decision process is unknown. We don't know that he announces that one came up heads whenever that is true. He might, for example, announce that one came up tails whenever that is true, and only announce that one came up heads when they're both heads.

We can gain some information by watching what the flipper does. If he repeatedly flips the coins, and about 3/4 of the time announces that one came up heads, then the classic analysis applies. However, if he only makes the announcement at some rate lower than 3/4, then something else is going on.

Since the problem statement showed us one flip without any other context, we haven't a clue as to what's really going on.

Jim Lyon
Sunday, June 23, 2002

What the flipper could have said or did say on previous flips is irrelevant.  What the flipper will do or might do on subsequent flips is equally meaningless.

The only thing that matters is what the flipper said on THIS event.  The flipper said that one coin is heads.  The only thing you can logically conclude from that is you didn't get two tails.

In common conversation, "One of the coins is heads" would mean that ONLY one coin is head, but not in a logic/probability problem.

Chad Hulbert
Tuesday, June 25, 2002

To demonstrate why context matters:

You're in a room watching someone repeated flip coins. Most of the time, after flipping them, he announces that one of the coins is tails. Only rarely does he announce that one of them is heads. After the announcement, he displays the coins. After several iterations, you deduce that he only announces that one is heads when they're both heads.

After you've made your deductions, I enter the room. I see the person flip two coins and announce that one of them is heads. What's the probability that they're both heads?

The paragraph above is exactly the stated problem. Do you see now why the algorithm that the person uses to make his announcement affects the answer?

Jim Lyon
Tuesday, June 25, 2002

You might also suggest the flipper lies about the whole situation and draws a response at random then tells you the "outcome."  Thus the problem is unsolveable.

In the situation you described, you're essentially asking, "What if 'one coin is heads' is really code for 'they're both heads'?"  Not only does that make the problem unsolveable, but uninteresting.  What if it's code for 'both coins are tails' or 'only one coin is heads' or 'there are no coins, you fool'?

Chad Hulbert
Wednesday, June 26, 2002

I'm afraid you missed the point again. By convention in these puzzles, all of the statements must be true. Also by convention, one need not make every possible true statement.

You say that this makes the problem unsolveable -- this is *exactly* my point.

To understand the difference, look at the original problem statement, and a couple of alternatives:

(Original) You flip two coins, and announce that one is heads. What's the probability that they're both heads?

(Alternative 1) You flip two coins. I ask "Is one heads?" You reply yes. What's the probability that they're both heads?

(Alternative 2) You flip two coins, and then, as you always do when they're not both tails, you announce that one is heads? What's the probability that they're both heads?

Note that both alternatives have well-defined answers. In order to believe that the original is solvable, you need to conclude that the phrase "as you always do when they're not both tails" was somehow implied in the problem statement. I decline to reach such a conclusion. Without it, the problem as originally stated is unsolvable.
Now, let's look at what could be done to the problem statement to

Jim Lyon
Thursday, June 27, 2002

Ignore the last line of the previous post. It was an editting mistake.

Jim Lyon
Friday, June 28, 2002

Agreed with the original point, that in language "one is heads" implies one and only one, while the problem itself does not.

However to consider the state of mind of the tosser, or what he 'really means', seems to be just trying to find faults with the problem.

If we knew the tosser announced about tails were any present, that would be a different problem. It's outside the question's scope, adding an extra layer of implication to his statement that is possible but which isn't reasonably suggested here.

The question tells us that he has announced "one is heads" and asks about the probability of heads for the other coin. It is not unreasonable to assume from this that his statement is made without a hidden motive or psychological history. We know we have one head, and there is probably some possibility of having a second head, otherwise why ask? That possibility is only affected by the constraints of his statement.

If you must, then think of it as a computer. It flips two coins and turns on a light if there is a head present. If the light turns on after a flip, what's the probability of having two heads?

Gary Who
Thursday, July 25, 2002

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