## Two Coins Puzzle

Browsing the Internet, I stumbled upon a coin puzzle which I slightly shrank to emphasize my point:

Carl flipped two coins and was asked if at least one of the two coins landed “heads up”. He replied, “Yes. In fact the first coin I flipped landed heads up.” What is the chance that Carl’s coins both landed heads up?

The standard answer is 1/2, because there are only two possibilities for the coin flips: HH and HT. But how do we know that these possibilities are equally probable?

The answer depends on what we expect Carl to say when he flips two heads. My personal assumption is that Carl is a perfectionist and always volunteers extra information. If Carl gets two heads, I would expect him to say, “Yes. In fact both coins I flipped landed heads up.” In this case the answer to the puzzle is 0.

Another strange but reasonable assumption is that upon flipping two heads, there is an equal probability that Carl would say either, “Yes. In fact the first coin I flipped landed heads up;” or, “Yes. In fact the second coin I flipped landed heads up.” In this case, the answer to the puzzle is 1/3.

I could describe an assumption for Carl’s answering strategy that leads to the puzzle’s answer of 1/2, but it looks too artificial to me.

This puzzle is not well-defined, but unfortunately there are many versions of it floating around the Internet with incorrect solutions.

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1. #### A:

“The standard answer is 1/2, because there are only two possibilities for the coin flips: HH and HT. But how do we know that these possibilities are equally probable?”
Simply use of basic statistics.

2. #### Tanya Khovanova:

Right after the flip, HH and HT are equally probable. The problem is that we need to find conditional probabilities of HH and HT given Carl’s statement.

3. #### Tanya Khovanova:

This comment was sent to me by George Zeliger, and I agree with it:

This is how I propose to edit the text. Instead of

“If Carl gets two heads, I would expect him to say, “Yes. In fact both coins I flipped landed heads up.” In this case the answer to the puzzle is 0.”

I would write:

If Carl did get two heads, I would expect him to say, “Yes. In fact both coins I flipped landed heads up.” Since he only mentioned the first coin in the puzzle, not both of them, the answer to the puzzle should be 0 – the second coin could not possibly land heads up, otherwise Carl would have reported that.

4. #### Christ Schlacta:

We can assume nothing about carl, and given that he accidentally answered a little more information than was requested is irrelevant. the probability of the second coin is still 1/2. Consider the extension of this problem: Carl was asked to flip a coin n times and was asked if at least one of the coins landed “heads up”. He replied, “Yes. In fact the first coin I flipped landed heads up.” What is the chance that Carl’s n/2th coin landed heads up?

5. #### Tanya Khovanova’s Math Blog » Blog Archive » Mr. Jones:

[…] some sense I didn’t forget about Mr. Jones. I wrote about him implicitly in my essay Two Coins Puzzle. His name was Carl and he had two coins instead of two […]